IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 24, NO. 3, MARCH 2009
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Resonant Contactless Energy Transfer With Improved Efficiency Stanimir Valtchev, Senior Member, IEEE, Beatriz Borges, Senior Member, IEEE, Kostadin Brandisky, Member, IEEE, and J. Ben Klaassens
Abstract—This paper describes the theoretical and experimental results achieved in optimizing the application of the series loaded series resonant converter for contactless energy transfer. The main goal of this work is to define the power stage operation mode that guarantees the highest possible efficiency. The results suggest a method to select the physical parameters (operation frequency, characteristic impedance, transformer ratio, etc.) to achieve that efficiency improvement. The research clarifies also the effects of the physical separation between both halves of the ferromagnetic core on the characteristics of the transformer. It is shown that for practical values of the separation distance, the leakage inductance, being part of the resonant inductor, remains almost unchanged. Nevertheless, the current distribution between the primary and the secondary windings changes significantly due to the large variation of the magnetizing inductance. An approximation in the circuit analysis permits to obtain more rapidly the changing values of the converter parameters. The analysis results in a set of equations which solutions are presented graphically. The graphics show a shift of the best efficiency operation zone, compared to the converter with an ideally coupled transformer. Experimental results are presented confirming that expected tendency. Index Terms—DC-DC power conversion, power conversion, resonant power conversion.
I. INTRODUCTION
E
NERGY converters acquired nowadays highly improved performance, i.e., higher switching frequency, higher power density, improved safety, smoother operation and higher reliability, and last but not least, a maximized efficiency for the whole range of operation. Although the efficiency is not always shown explicitly in the publications on new topologies, as a general rule the best energy converter is the most efficient one. The efficiency is one important reason why the resonant and especially, series loaded series resonant (SLSR) power converters are constantly gaining popularity. Their capabilities for Manuscript received January 28, 2008; revised May 30, 2008. Current version published April 08, 2009. Recommended for publication by Associate Editor B. Ferreira. S. Valtchev was with the Instituto de Telecomunicações-IST, 1049-001 Lisbon, Portugal. He is currently with the Faculdade de Ciências e Tecnologia-Universidade Nova de Lisboa, Quinta Torre, 2829-516 Caparica, Portugal (e-mail:
[email protected]). B. Borges is with the Instituto de Telecomunicações-IST, 1049-001 Lisbon, Portugal. She is also with the Instituto Superior Têcnico-Universidade Técnica de Lisboa, 1049-001 Lisbon, Portugal. K. Brandisky is with the Technical University of Sofia, 1000 Sofia, Bulgaria. J. B. Klaassens is with the Delft University of Technology, Faculty of Electrical Engineering, 2628CD, Delft, the Netherlands. 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.2008.2003188
soft-switching, high-frequency operation, minimized volume and weight, result in low maintenance cost, high reliability and highest efficiency. There are still arguments against the production of this class of converters, due to: (1) the existence of loops of dissimilar dynamics (one or more resonant loops and the input and output filters) provokes an increased complexity in converter design and dimensioning, not permitting to predict the waveforms of the electrical variables as easy as in the case of the hard-switching converters; (2) as a consequence of (1) the control strategy and the inherent stability criteria of the system are more complex, thus not allowing simple two-loop stabilizing methods, as for example the current mode control; (3) although it is possible to nearly cancel the commutation losses in resonant converters, it has to be admitted that higher conduction losses may occur, mostly when the current form factor is much larger than the value of one. In general, it is possible to overcome the above mentioned difficulties, as well as some other problems, specific for resonant power conversion. If the drawbacks are minimized, the great capabilities of the resonant converters categorize them as the best or even the only solution for specific applications, either for low or for medium power levels. One application field, where the resonant converters are best suited is contactless energy transfer. The widely used method, in different configurations [1]–[5], is the Inductively coupled power transfer (ICPT) applied between magnetic coils at a relatively high frequency. The presence of resonant processes in this case is unavoidable, not only because of the intrinsic reactive components involved in the inductive link, but also because additional capacitors are intentionally included to compensate for the low coupling coefficient. Considering that the resonant processes are intrinsic to these circuits, the knowledge of the resonant conversion becomes not only extremely advantageous but also indispensable. The problem stated here is to discover and analyze the influence of the loosely coupled transformer on the SLSR power converter performance, optimizing it for the best possible efficiency. The objective of this paper is to propose a new method for modelling and analysis of the SLSR converter used for contactless applications. The model consists of two separate equivalent circuits derived from the complete (i.e., more complex) equivalent circuit. The equations characterizing them are mathematically forced to yield instantaneous values of the physical variables that will be identical to those produced by the original circuit. Those two equivalent circuits represent the approximated behaviour of the primary and the secondary sides of the converter. The approximation’s aim is to obtain structures of the circuits similar to the idealized equivalent circuit considered in
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[6] and [7]. Through this approach, the ideal converter analysis can be directly applied, making it easier to obtain the converter output characteristics, thus allowing a faster evaluation of the converter performance. Due to the circuit complexity, this approach is more effective to allow a rapid evaluation of the converter variables for large variations of the operation regimes. As a consequence, it becomes easier to build a graphical presentation of the characteristics showing the areas of the highest efficiency and to apply this knowledge in the design of efficient contactless converters.
Normalized output voltage Equivalent normalized output voltage (Thevenin). Equivalent loss resistance of the circuit. Load resistor. Air gap distance. Voltage between nodes A and B. Voltage on the resonant capacitor. Amplitude of the resonant capacitor voltage.
II. LIST OF SYMBOLS AND ABBREVIATIONS
Primary approximated amplitude of Multiplying factor.
Secondary approximated amplitude of (imaginary).
Output filter capacitor.
Resonant loop excitation voltage.
Resonant capacitor.
Equivalent output voltage (Thevenin).
DC output voltage.
Transformed output voltage.
DC input voltage. Total energy stored in the LC tank.
Approximated output voltage values, version “ ” and “ ”
Normalized switching frequency
Dimensionless (angular) time variable
Resonant frequency.
Angular instant of transistor commutation.
Switching frequency.
Dimensionless (angular) half switching period.
Current in the secondary winding of the power transformer.
,
,
Imaginary impedance channels. Characteristic impedance of the LC loop.
Current in the resonant inductor (primary).
Characteristic impedance of the primary equivalent LC loop.
Magnetizing inductance current.
Characteristic impedance of the secondary equivalent LC loop.
Average value of the output current. RMS value of the resonant current.
Efficiency.
Virtual amplitude of the first segment of the resonant current.
Normalized first time interval:
Virtual amplitude of the second segment of the resonant current.
Normalized first time interval: primary circuit.
in
Magnetic coupling factor.
Normalized first time interval: secondary circuit.
in
Total inductance of the primary winding.
Normalized second time interval:
Total inductance of the secondary winding.
Normalized second time interval: primary circuit.
in
Normalized second time interval: secondary circuit.
in
Total primary inductance. Total secondary inductance.
Current form factor.
Magnetizing inductance. (Additional) resonant inductor.
Leakage inductance of the primary winding.
FM
Frequency mode (of operation).
Leakage inductance of the secondary winding.
ICPT
Inductively coupled power transfer (converter).
Equivalent inductance (Thevenin).
PWM
Pulsewidth mode (of operation).
Mutual inductance.
SLSR
Series loaded series resonant (converter).
Transformer ratio.
ZVS
Zero voltage switching.
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Fig. 1. Basic circuit of an ideal SLSR dc-dc power converter: a) basic power circuit; b) equivalent circuit; c) typical state variables waveforms.
III. IDEALLY COUPLED SLSR CONVERTER A generic power circuit of the ideal SLSR converter is presented in Fig. 1(a). One of the possible modes of operation is and characterized by alternate closing the pairs of switches (respectively, and ) at a frequency above the resonant frequency, i.e., in a super-resonant mode. Other techniques of switching can be selected for the same topology and some variations of the topology are known as well, but in general, the circuit of Fig. 1 represents the most important idea: to guarantee zero voltage switching (ZVS) for all possible modes of operation. For the non-ideal SLSR converter with a loosely coupled transformer, it would be convenient to apply the normalized notation used in [6] and [7], where an ideal transformer is assumed in the analysis. The normalization of voltages, currents, frequency, etc. serves well the aim of obtaining generalized expressions that can be used to describe any specific converter. Applying the same notations for converters with ideal and non-ideal transformers makes it easy to compare their operation. The analysis of the ideal converter will result in the calculaas illustrated by the equivtion of the state variables, and alent circuit in Fig. 1(b). Providing that the output capacitor is sufficient to maintain a constant output voltage during at least one switching period, the load can be replaced by an ideal . The output rectifier diodes and will voltage source conduct the positive and negative half-waves of the resonant current, respectively, imposing at the transformer primary terminals an alternating square wave voltage , which polarity always opposes the direction of the resonant current. The amplitude of is equal to the transformed voltage , where the voltage is the transformer ratio. The operation of the idealized SLSR converter is, therefore, equivalent to the excitation of an LC circuit by the combination of two alternating square voltages: and , being out of phase. The voltage is generated by
, , and and its amplithe action of the switches . Together with the voltage tude is equal to the input voltage , the voltage is presented in Fig. 1(b). The excitation for the LC resonant circuit is formed by the addivoltage tion of the voltage sources and and hence, during the switching period this voltage assumes consequently the values: , , or . In the frequency mode (FM) of regulation, the current in the resonant LC circuit is continuous. This means that during the four different time intervals of operation, corresponding to the four different , the current in the LC loop values of the excitation voltage has no interruption. The normalized notation (e.g., the index ) is omitted for simplicity. This requires that all the voltages are . So the normalfrom now on, divided by the input voltage . The normalized output voltage is denoted as ization of the currents is done by multiplying their real value , where is the characteristic impedance . by Then the state-variable equations are given by the matrix equation:
(1) In this expression the time variable is also normalized, i.e., . The general solution, for any time interval, that starts , will be: at the initial point of time:
(2) (3) As the cyclic stability of the operation implies and as shown in Fig. 1(c), the variables of the resonant circuit demonstrate a
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symmetry: . This symmetry results in values that are and of for the first half switching period and and in the second half switching period, allowing to consider only the first half of the switching period, and its segments defined by the switching interdenoted as and shown in Fig. 1(c). The two excitation vals voltages and will have values corresponding to the . The initial conditions are also first switching half period symmetrical, defined by the maximum voltage of the resonant capacitor when the resonant current crosses zero: Fig. 2. Output characteristics q and idealized transformer.
= f (I ) for different current form factors
(4) , transferred to and from the Because the total energy resonant circuit by the excitation voltages during the is equal to zero, (5) is added: half switching period
(5)
and from The obtained values for the angular intervals (8) and (9), together with the calculated amplitudes (10) and (11) are applied for calculating the characteristic waveforms of the idealized SLSR super-resonant converter. The waveforms are then calculated in many points of operation and the output characteristics are obtained. A. Output Characteristics
The general solution for the two sub-intervals can now be written as:
(6)
(7) Those equations are valid for any continuous-current mode of operation (sub-resonant or super-resonant) and any possible excitation voltage combinations. In case of applying a super-resonant operation of the SLSR converter, for the time interval , the normalized excitation voltage are and in the next interval , . The normalized values of the time intervals are denoted as and [see (6) and (7)] and presented in normalized form:
(8) (9) is the absolute value of the peak values The voltage of the normalized resonant capacitor voltage. The (virtual) amplitudes of the resonant current sinusoidal segments, following (3) are: (10) (11)
It was shown in [6] that the efficiency achieved by the SLSR converter depends on the shape of the resonant current, as shown in (12):
(12) compares the where the current form factor value of the current in the resonant circuit to the average value of the rectified resonant current (equal to the output current of the converter) . In (12), the efficiency, denoted by is expressed through (the equivalent loss resistance of the (the load resistance). circuit) and The output characteristics presented in Fig. 2 are plotted for different values of , thus showing the most desirable operation zones (the zones with the minimum value of the current form factor). The current form factor curves, when corrected by the knowledge about the maximum output power (a set of hyperbolic curves) show that the expected highest obtainable efficiency is placed in the upper left area of Fig. 2. The output characteristics in Fig. 3 illustrate the operation at a number of fixed switching frequencies and thus indicate the position of the best operation zone in case the FM of regulation is applied. Considering the above information, the converter operation with an ideally coupled transformer is recommended in the zone of a normalized output voltage higher than 0.6 and the normalized output currents up to 2.0. The expected switching frequency, that suits best the maximum output power transfer, up to . The FM of operation is is from suitable for the highest levels of output power, requiring the highest efficiency. In case of a lower demand of output power, the pulsewidth mode (PWM) operation with a relatively lower efficiency is an alternative investigated in [7]. To operate in the
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Fig. 4. Loosely coupled SLSR converter, equivalent circuit.
Fig. 3. Output characteristics for fixed normalized switching frequency F and idealized transformer.
required zone it would be necessary to adjust the transformer , ratio , to choose a suitable normalized frequency and to provide the characteristic impedance that guarantees the necessary normalized values. The inclusion of a loosely coupled transformer into the SLSR circuit is expected to provoke displacement of the zones of best operation. The study of the non-ideal transformer will be based on the transformer model which elements are expected to vary with the magnetic coupling. To obtain a faster and easier calculation of the output characteristics, the total equivalent circuit will be split in two ideal resonant circuits with recalculated components. This operation allows applying the knowledge about the ideally coupled converter, described in [6] and [7], maintaining a good approximation to the full equivalent circuit results.
IV. SLSR CONVERTER OPERATION WITH NON-IDEAL TRANSFORMER The SLSR power converter applied for ICPT (i.e., contactless energy transfer) requires a sufficient air gap distance between the primary and secondary halves of its magnetic link. This non-ideal coupling is characterized by an increased reluctance of the total magnetic circuit. As a result, the transformer model (“Steinmetz” circuit) illustrated in Fig. 4, presents a lower value . This lower value of the equivalent magnetizing inductance provides an alternative path for the resonant current, and thus reduces the current transferred to the secondary, especially when the output voltage is high. According to [6], the ideally coupled converter has its highest efficiency (the value of has its minimum) exactly in the highest (normalized) output voltage zone, as shown in Fig. 2. When the magnetic coupling is becoming weaker, the parameters of the loosely coupled transformer are expected to be significantly different from the ideal ones and to influence the operation of the SLSR converter.
A. Magnetic Modelling To identify which parameter will influence the efficiency of the power converter, magnetic simulations and measurements
Fig. 5. Three-dimensional simulation arrangement (MagNet program) for pot core and planar core.
were carried out. The simulations were executed both in FEMM [10] and in MagNet [11] programs. In Fig. 5 the basic arrangements for the MagNet simulation are shown, by examples of pot core and to planar core ferromagnetic sets. The results, shown further on are obtained from the pot core (in the planar case they are comparable). These two magnetic link constructions are widely applied: the pot-core is similar to the implementations of a rotary joint and the planar core may represent the charger of certain electric vehicles (e.g., the transmission is from the floor). As an example, two different FEMM simulations are shown in Figs. 6(a) and 6(b). Fig. 6(a) illustrates the behaviour of an imaginary ferrite core with linear characteristics, i.e., possessing a constant permeability. In Fig. 6(b) a more realistic, non-linear ferrite material is represented with characteristics of a standard grade N27 material. Both figures show almost equal magnetic field distribution and intensity. The calculated from the simulations values of the leakage inductance and the magnetizing inductance, are shown in Figs. 7(a) and 7(b), representing the linear and non-linear magnetic characteristics, correspondingly. The simulated values are confirmed by measuring results, demonstrated in Table I. The conclusion is that the largest variation is observed in the value of the magnetizing inductance. It is rapidly decreasing with the growing air gap distance between the two transformer halves, while the leakage inductance value is keeping a nearly constant value, especially when the air gap distance between the magnetic core halves is larger than the minimum. To reduce further the effect of the leakage induc(similar to the tance variations, a fixed external inductance resonant inductance in an ideal SLSR converter) is connected in series with the transformer. This additional fixed inductance value is later included in the calculations as a multiple of the . leakage inductance value, i.e.,
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Fig. 6. Simulation by FEMM program, pot core PM62/49, air gap distance s ferrite characteristics.
= 2:6 mm: a) linear magnetic characteristics, = 1400; b) non-linear N27 grade
( ) and leakage inductance (L ) for different distance s between the ferrite halves: a) linear magnetic characteristics; b) non
Fig. 7. Magnetizing inductance L linear magnetic characteristics.
TABLE I MEASURED INDUCTANCES, DEPENDING ON AIR GAP DISTANCE (PM62/49 POT CORE OF N27 GRADE, 20 TURNS)
B. Electrical Circuit Analysis The contactless super-resonant SLSR power converter, due to its reduced magnetizing inductance, has two different “reso-
and the secondary , as shown nant” currents: the primary in Fig. 4. These two currents differ by the considerable value and in consequence, and of the magnetizing current are displaced in phase, as presented in Fig. 8. The current is still in phase with the output rectifier voltage but the primary current waveform is significantly delayed. Similarly to the ideal SLSR converter case, the division of the converter operation in time intervals made in Fig. 8 reflects the charging process of the resonant capacitor. The normalized time denoted by “0” corresponds to the negative maximum of the resonant capacitor voltage and the normalized time “ ” denotes the time of its positive maximum. At both instants the resonant current assumes zero values. The half is thus divided in two intervals: switching period and but in fact inside the second time interval the current crosses the zero value, inverting the polarity of the output source and changing the form of the equations. The equations become even more complex when the input is applied in discontinuous mode, i.e., in PWM voltage or zero output current intervals exist. These (reduced output power) modes of operation will not be considered as the aim
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Fig. 8. Loosely coupled SLSR converter: typical state variables waveforms.
here is to achieve the highest possible converter efficiency at high output power. Even for the FM operation, the resonant variables are defined through several differential equations, different for each loop and for each time interval. The structure and the initial conditions of each equation, depends on the process development during the previous interval. This set of differential equations is unfortunately becoming too heavy to obtain its simple solution. There are publications showing examples of analytical solutions even in multi-interval (piece-wise) correct equations, e.g., [8], which solutions are not easy to obtain, neither fast calculable (usually in implicit form). In the case of low magnetic coupling the SLSR converter has up to five intervals in each switching half-period [9], which produces a high order system of differential equations. The equations of the SLSR converter with an ideal transformer offer rapid solutions for many different changing circuit parameters, facilitating the decisions for the strategies of regulation. The simplicity of the ideal case equations, lead to the idea of dividing the model of Fig. 4 into two separate sub-models, each corresponding to one of the (modified) transformer sides. By this division, the order of the equations is lowered, permitting a faster and easier (approximate) solution, capable to be used for the analysis and the regulation of the SLSR converter with non-ideal transformer. The two (imaginary) sub-circuits must have their parameters recalculated, in order to obtain two structures, identical to the circuit shown in Fig. 1(b). Therefore, each set of equations must be similar to (1)–(11). The shapes of and , shown in Fig. 8, are equal the two resonant currents to the ideal waveform presented in Fig. 1(c), confirming by this the idea of two ideal sub-circuits. 1) Primary Side Circuit Model: Considering a cyclic stable operation and ignoring the internal resistive losses, the circuit in Fig. 4 shows no dc component in its electrical variables, seeing that: is symmetrically alternated each — the primary voltage half switching period ;
— the primary current is free of a dc component due to that symmetry and the presence of the capacitor ; — the secondary voltage is also symmetrical because the rectifier is opposing to the secondary current the same output voltage (constant during a number of switching periods); — the secondary current is symmetrical as the rectifier opand poses to it the symmetrically alternating voltage for cyclic stability the charging and the discharging of the (lossless) reactive elements is equal during each period; — the harmonic content of the current in the magnetizing inhas no dc component because it is a sum of ductance the other two purely ac currents; — the circuit in Fig. 4 consists exclusively of reactive ele, , , etc. ments with no losses, i.e., , When the above conditions are considered to be valid, the circuit from Fig. 4 permits to be redrawn as an equivalent circuit that is (approximately) correct in relation to the primary side (primary current) of the converter. The modification of the circuit applies the Thevenin’s transformation, operating with purely reactive elements. To generalize the relation between the secondary and the primary sides the transformer ratio is adopted further on. The result is that the magnetizing inductance , seen from the primary or seen from the secondary (and the mutual inductance , as well) will present equal values. The and primary and the secondary leakage inductances are also regarded as symmetrical, i.e., . The applied simplification is not limiting the conclusions and permits to recalculate the realistic values by application of the real transformer ratio . The correctness of the above simplifications is of course, limited, and concerns especially the converter’s cyclic stabile mode of operation. Regarded from the primary current loop, the equivalent reacand ) result into (indexed tance (pure inductances “T” as Thevenin):
(13)
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Fig. 10. SLSR converter secondary side modelling: a) splitting in two impedance channels; b) recalculated sub-circuit for the secondary.
Fig. 9. Primary equivalent circuit of the SLSR dc-dc converter with a loosely coupled transformer.
The ratio is the classical magnetic coupling coefficient (it remains equal for the primary and for the secondary sides of the and is transformer, since the symmetry adopted here):
to be calculated too, in order to be compared to the secondary side parameters and to achieve the necessary synchronization. Altogether, the (1)–(11) must be rewritten with the recalculated , , , , etc. when applied to the priparameters mary side equivalent circuit, presented in Fig. 9, with the index 1 to denote their validity for the primary circuit. Finally, the normalized values for the forward conduction and for the reverse (diode) conduction (transistor) interval of the primary current are recalculated in (18) and interval (19).
(18) (19)
(14) For simplification the additional (fixed value) resonant inducis chosen proportionally related to the almost constant tance : leakage inductance (15) In this case the total primary inductance
is:
(16) The output voltage amplitude (Thevenin’s transformation) applying a classical inductive voltage diis calculated from vider and a pure ac voltage source:
(17) In Fig. 9 is shown the circuit model used for solving the primary side equations. The equations that describe the circuit in Fig. 9, have the same form as the ones related to the ideal SLSR converter and presented in [6] as the structure of the circuit is equal. It is necessary to apply the recalculated above values of the output voltage and the total inductance in the resonant loop. The values of the charand the normalized time need acteristic impedance
The voltage is the absolute value of the resonant capacitor voltage peaks: or , referred to Fig. 9. These voltage peaks correspond to the resonant current zero and . crossings in 2) Secondary Side Circuit Model: The model used to calculate the primary current is not suitable for the secondary current because only a limited fraction of the primary current reaches the secondary, due to the weak magnetic coupling of the transformer. The secondary side must be modelled by another idealized circuit (Fig. 10), which structure is based on the presumption that the fraction of the primary resonant current that reaches the secondary is proportional to the magnetic coupling coefficient . To achieve this fractioning effect, the primary impedance is split in two channels, both originated from the , as illustrated in Fig. 10(a). The upper input voltage source channel in Fig. 10(a) consists of the three primary impedances divided by the coupling factor , while the lower channel is a . connection of the same primary impedances, divided by One of the channels is directed to the output, supplying the secondary current . The other one is connected to the magnetizing inductance and thus, produces no current in the secondary. The output voltage is generally not equal to , and has an extra index “ ”. If reconnected, the two split parts of each original reactive , .A component must result in its original value, i.e., , pair of imaginary characteristic impedances of the two primary and . The two impedances channels can be denoted as , as shown in are related in the same proportion (20), identical to that of the individual component impedances.
VALTCHEV et al.: RESONANT CONTACTLESS ENERGY TRANSFER WITH IMPROVED EFFICIENCY
If joined again the two split channels, the primary characteristic . impedance results in the original value
(20)
In the secondary circuit model (Fig. 10), the secondary maintains its original value as it is not leakage inductance subjected to splitting. The transformer ratio continues to be in the final circuit, as shown in Fig. 10(b). The considered and are thus increasing, being divided by values of (which value is between 0 and 1). The value of is decreasing being multiplied by the same coefficient: the magnetic coupling . The total inductance in the circuit of Fig. 10(b) is equal to in (21), considering again and equal the sum , which in turn is equal to as adopted in (15). to
(21) of the total second equivaThe characteristic impedance ), shown in Fig. 10(b) is compared in (22) lent circuit (incl. to the corresponding impedance of the primary equivalent circuit, illustrated by Fig. 9. The relation between the two circuits is important in order to achieve a synchronous functioning of the two circuits expressed in their equations.
(22) The resonant frequency of the circuit in Fig. 10(b) remains the same as the resonant frequency of the circuit in Fig. 9, because the capacitance and the inductance are multiplied and divided correspondingly, by an identical value of , as illustrated in (23).
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and falling slope of the resonant current (i.e., between in Fig. 1) will be much steeper when the output voltage is increased. Inspected by that previous knowledge, the waveform of the secondary current , presented in Fig. 8, is similar to the one produced by an ideally coupled SLSR converter, which involves an output voltage higher than the real voltage present at the output. In addition, the weak coupling of the transformer makes difficult to transmit current to the output, in cases when the output voltage is too high. This statement can be confirmed by the circuit in Fig. 4, where the (ac) voltage applied to the primary of the transformer is diand and presents a vided by the series connection of proportion of voltage to the secondary side. It would be difficult to supply current to a normalized output voltage source , (approximately). It is the same to higher than the value of consider the normalized output voltage as higher than the real . This recalculated value is closer to the reality, when the operation is far enough from the resonant frequency and the normalized output voltage is relatively high. When the normalized and especially when the switching frequency is closer to is low, the above mentioned dividing of the output voltage input voltage is not observed, because the resonant circuit provides a very low impedance to the input of the transformer. In this case the output voltage can be considered unchanged, . In order to take into consideration the two extreme cases, the (i.e., the square wave with amplitude ) output voltage from Figs. 4 and 10, is substituted by two extreme values, and . The first recalculated normalized denoted as output voltage of the secondary equivalent circuit corresponds to the operation at a switching frequency far from the resonance and (or) at high output voltage. This situation (as it was mentioned above) elevates effectively the normalized output voltage . The second extreme value reflects to the opposite situation (i.e., low output voltage, close to the resonant frequency operation, or both of them) and maintains . The mean value the original output voltage value, of the two extreme values is adopted in the expression (24) as the best general approximation and is verified by calculating different combinations of realistic modes of SLSR converter operation.
(24)
(23) By substitution of and in (22) and (23), these expressions assume values correspondent to the converter with an ideal transformer, i.e., and . The above mentioned adjustment of the impedances is not sufficient to represent the secondary current by the current of an ideal SLSR converter. It was demonstrated in [6] and [7] and it can be calculated from the expressions (8)–(11) that the
is included into the secThe recalculated voltage value ondary equivalent circuit of Fig. 10(b) that represents in an idealized form, the secondary current of the contactless SLSR converter. Together with the primary equivalent circuit of Fig. 9, these two circuits are substituted for the original circuit of Fig. 4, with its loosely coupled transformer. Although the expressions (13)–(24) serve as the basis for the combined and synchronous functioning of the model, some more adjustments are needed in order to correlate the calculated variables with the real circuit. The secondary circuit, shown in Fig. 4 has no separate resonant capacitor, and thus the introduced virtual capacitor in Fig. 10(b) will need a newly defined peak voltage , . The different from the defined in (18) and (19) voltage
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Fig. 11. Simulation results from the total equivalent circuit (SLSR contactless converter) and from the two approximated sub-circuits at K : and q : : upper plot: secondary currents; lower plot: primary currents.
=09
= 0 25
expressions (25), (26) present the new, different conduction intervals for the circuit of Fig. 10(b).
(25) (26) As the switching frequency must be identical for the primary and the secondary idealized circuits, the normalized half is expressed in (27). switching period
(27) is calculated The switching half period by (18) and (19), for the specified output voltage and for the of the resonant capacitor voltage. In fact, the peak value of the resonant capacitor in Fig. 9 approximates voltage well the real resonant capacitor voltage. Knowing that the value is directly proportional to the average value of the rectified resonant current, as shown in [6] and applying (27), the variable values in the secondary circuit, are also obtained. The resonant capacitor in the secondary is only virtual, artificially introduced in Fig. 10(b) and its value does not correspond to the original capacitor of the circuit in Fig. 4. By the calculation, the synchronous functioning is acquired, between the circuits from Fig. 9 and Fig. 10(b). The validity of the described above approximation is demonstrated graphically, in a couple of different operation modes, presented in Figs. 11–13. These figures show the time diagrams
F
= 1 44, = 1, :
a
of the currents in the primary (bottom diagrams) and in the secand are ondary (top diagrams). The currents denoted by simulated applying the original circuit, presented in Fig. 4. The currents with identical names but indexed as “approx” are produced by the models, presented in Fig. 9 and Fig. 10(b), with their recalculated values, corresponding to (13)–(27). The plots in Figs. 11 and 12 are taken at different switching . This equals about frequency and at the same coupling, 1 mm distance of the two halves of PM62/49 ferrite core and is completely sufficient mechanically even for rotary connection. The circuit has additional resonant inductance, equal to the . As it will be seen, there is no visual leakage one, i.e., difference between the behaviour of the real and approximated circuits. The current in the secondary is lower (about 20% less amplitude and slightly different shape) and this also perfectly corresponds to the expected waveforms. and the lower In Fig. 12, the lower frequency, produce waveforms closer to full sinuoutput voltage, soids, almost with equal amplitudes between primary and secondary (close to the resonance) and again with excellent coinciding between the models. It can be seen that the low voltage of the output helps to overcome the problems of the loosely coupling, transferring more current to the secondary and also confirming the approximated models. The currents in Fig. 13 correspond to a higher output voltage, . As a consequence, the difference between the prii.e., mary and the secondary current is slightly higher but the accuracy of the models is still quite good. The accuracy of the model is verified due to the insignificant difference between the approximated and the real current waveforms. Based on the recalculated values from (13)–(27), the output characteristics of the contactless SLSR converter can be estimated with good accuracy. In Fig. 14 the FM regulation char-
VALTCHEV et al.: RESONANT CONTACTLESS ENERGY TRANSFER WITH IMPROVED EFFICIENCY
Fig. 12. Simulation results from the total equivalent circuit (SLSR contactless converter) and from the two approximated sub-circuits at K : and q : : upper plot: secondary currents; lower plot: primary currents.
=09
=01
Fig. 13. Simulation results from the total equivalent circuit (SLSR contactless converter) and from the two approximated sub-circuits at K : and q : : upper plot: secondary currents; lower plot: primary currents.
=09
=05
acteristics are presented, i.e., the normalized output voltage in function of the normalized output current with the frequency as parameter. Fig. 14(a) illustrates the magnetic coupling coefficient , which corresponds to an approximate disbetween two halves of PM62/49 ferrite cores. tance , i.e., to 7 mm distance. Fig. 14(b) corresponds to It can be observed in Fig. 14 that it is not possible to reach higher output voltages when the coupling coefficient is too low (practically, must be kept lower than , at least). The same
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F
F
= 1 152, = 1, :
a
= 1 152, = 1, :
a
limitation of the output voltage is observed in the output characteristics of Fig. 15, drawn when the current form factor is taken as parameter. In order to reflect better the real output values, the current form factor in this figure is modified, invalue of the primary current and the average volving the value of the secondary current, as defined in (28). (28)
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Fig. 14. Normalized output characteristics of the contactless power converter at a constant switching frequency F and: a) for PM62/49) and ; b) for PM62/49) and (i.e., distance .
a = 1 K = 0 :5
q
s = 7 mm
a=1
K = 0:8 (i.e., distance s = 2:5 mm
Fig. 15. Normalized output voltage in function of the normalized average output current for constant values of the current form factor (low leakage inductance); b) and .
a = 0:01
K = 0 :8
a=1
Because of the decreasing value of the current in the secondary even at slight decrease of the magnetic coupling, the current form factor in (28) shows its most favourable values to move to the right of the output characteristics, i.e., closer to the higher values of the normalized output current. This can be seen even . This movement of in Fig. 15(a) where the coupling is the best operation zone is due to the best power transfer when the switching frequency is closer to the resonance (the impedance in Fig. 4, is becoming branch, represented by the inductance less important). Fig. 15(b) shows a further development of the output characteristics, when the magnetic coupling becomes really weak. In Fig. 15(b) a notable shift of the expected best efficiency zone is seen, both to the right (high output current) and to the bottom (low normalized output voltage). This move is in fact, again in the direction of the resonance. The above statement can be confirmed by the modified regulation characteristics, presented in Fig. 16. In this figure the normalized output voltage is plotted in function of the switching frequency, for different fixed values of the current form factor . In Fig. 16(a), where the magnetic coupling is still close to the ideal, the best operation zone is very close to the ideal conand higher verter one, i.e., switching frequency up to , the best zone is output voltages. In Fig. 16(b), where moved down to the lower output voltage and the operation fre. quency is much closer to the resonance The practical results show that the real shift of the best operation zone is slighter than the expected from Figs. 15 and 16. This
: a) K = 0:99 and
is due to the losses in the secondary (rectifier, windings, etc.), which relative role increases when the output voltage decreases. In addition, the secondary losses increase in their absolute value as well, because the production of an equal output power will necessitate a higher output current when the normalized output voltage decreases. In fact, to achieve the required value of the output voltage and to keep the advised lower normalized output voltage, the transformer ratio is recalculated. Although this adjustment alleviates the problem in the secondary and its rectifier, it transfers some difficulties to the primary, where the current must rise in proportion to the introduced correction of the transformer ratio. V. EXPERIMENTAL CONTACTLESS SLSR CONVERTER The 1 kW SLSR converter is specially prepared to be versatile and to be capable to operate in many possible regimes, either in the continuous current, i.e., frequency mode (FM) or in the discontinuous current regulation mode. The discontinuous mode of SLSR converter operation allows a deeper regulation of the minimum output power, as demonstrated in [7]. To reach the lowest output power is not a substantial problem of the efficient contactless power transfer and it will not be experimentally shown for this article and in the presented experimental results, FM regulation is exclusively applied. A photograph of the experimental converter is shown in Fig. 17. In Fig. 18(a) the primary and secondary currents of this converter are simulated, calculating the approximated equivalent
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Fig. 16. Normalized output voltage q in function of the normalized switching frequency F for constant values of the current form factor : a) K a : (low leakage inductance); b) K . : and a
= 0 01
= 08
=1
= 0:99 and
Fig. 17. Experimental 1 kW resonant converter: upper side: receiver; lower side: transmitter.
circuits for the same parameters as the applied in the experiment. For the experimental waveforms, presented in Fig. 18(b), the while the output voltage input voltage applied is is referred to the primary as (the ). The resotransformer ratio in this case is fixed at and ) but here nant frequency is changeable (by varying . The oscilloscope image shown in also is fixed: Fig. 18(b) presents the experimentally obtained waveforms of and and the currents in the primary and the voltages secondary windings. The slightly higher values of the secondary current that are calculated (compared to the measured values) is due to the not included in the calculation rectifier voltage drops (relatively high in this case). A selection of the measured data is presented in Fig. 19(a), , i.e., where the load voltage was fixed at , since . In Fig. 19(b) a fixed output current , and because of the limitations of was maintained . The the equipment, the input voltage is risen to output voltage is shown in its normalized form. The switching frequency variation is applied for regulation of the output power. In the version (a) the changing output variable is the current and in the case (b), the normalized output voltage is varying. Even with the restrictions imposed on the equipment, the figures
Fig. 18. Simulated waveforms (applying the approximated values) and exper: (distance s : for PM62/49), input imental waveforms for K voltage E , output voltage E (i.e., q : ,n ): , 50 V/div (secondary), 2 A/div (primary a) simulation, scales: 100 V/div v and referred secondary); b) experimental results, scales: 100 V/div v , 50 V/div (secondary), 2 A/div (primary), 10 A/div (secondary).
= 160 V
= 08 = 2 5 mm = 17 V = 0 53 = 5 : 1 ( ) ( )
show confirmation of the predicted tendency in displacing the maximum efficiency zone.
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VI. CONCLUSIONS As the leakage inductance keeps a stable value compared to the magnetizing inductance in the course of varying the distance between the transmitter and the receiver, it has been shown that the SLSR converter is a better choice when working with a loosely coupled transformer. The series compensation of that transformer reactance by a resonant capacitor makes the properties of the series resonant converter more predictable and reliable. In this case, the (series) resonant frequency remains almost unchanged with the variation of the distance transmitter/receiver. The presented method of analysis of the SLSR converter circuit with a loosely coupled transformer as a magnetic link was proved to be valid (at least for the most practical distances and magnetic coupling coefficients). The method allows applying the knowledge about the ideally coupled SLSR super-resonant converter to the more complex case of contactless energy transfer converters. The presented reallocation of the area of maximum efficiency is important because it permits a suggestion for the zone of operation at the maximum output power (where the efficiency is essential). The simplified analysis is also applicable to the implementation of efficient, faster and secure control strategies.
REFERENCES
Fig. 19. Efficiency measured at different distance s between the halves of PM62/49 and at a different load: a) input voltage E , output voltage E and s (curve 1), 1 mm (curve 2), 2 mm (curve 3); b) , fixed output current I and s input voltage E (curve 1), 2 mm (curve 2).
= 90 V
= 0 mm = 200 V
= 100 V = 2A
= 0 mm
Each curve in Fig. 19 represents a set of measured efficiency points that corresponds to a given fixed air gap distance between the transmitter and receiver half cores of ferrite. Comparing the curves, it is seen that in general, that the efficiency decreases when the distance increases. This efficiency variation is natural and unavoidable, but at each maintained distance between the transmitter and the receiver, the corresponding curve shows a point where the efficiency has its maximum. The position of that maximum point differs, depending on the distance . When the distance increases, the maximum point exhibits a shift to the right side of Fig. 19(a), i.e., to the higher output currents, while in Fig. 19(b) the maximum efficiency point moves to the left, i.e., to a lower normalized output voltage. The directions of those shifts (although not so accentuated as expected by the current form factor changes) are predicted by the analysis and confirm the viability of the approximations introduced to the contactless SLSR converter equivalent circuit.
[1] J. Hirai, T. Kim, and A. Kawamura, “Wireless transmission of power and information for cableless linear motor drive,” IEEE Trans. Power Electron., vol. 15, no. 1, pp. 21–27, Jan. 2000. [2] J. Hirai, T. Kim, and A. Kawamura, “Practical study on wireless transmission of power and information for autonomous decentralized manufacturing system,” IEEE Trans. Ind. Electron., vol. 46, no. 2, pp. 349–359, April 1999. [3] H. Jiang, G. Maggetto, and P. Lataire, “Steady state analysis of the series resonant DC-DC converter in conjunction with loosely coupled transformer—Above resonance operation,” IEEE Trans. Power Electron., vol. 14, no. 3, pp. 469–480, May 1999. [4] J. Barnard, J. Ferreira, and J. van Wyk, “Sliding transformers for linear contactless power delivery,” IEEE Trans. Ind. Electron., vol. 44, no. 6, pp. 774–779, Dec. 1997. [5] D. Pedder, A. Brown, and J. Skinner, “A contactless electrical energy transmission system,” IEEE Trans. Ind. Electron., vol. 46, no. 1, pp. 23–30, Feb. 1999. [6] S. Valtchev and J. B. Klaassens, “Efficient resonant power conversion,” IEEE Trans. Ind. Electron., vol. 37, no. 6, pp. 490–495, Dec. 1990. [7] S. Valtchev, “Some regulation characteristics of pulse-width modulated series resonant power conversion,” in Proc. 6th Conf. Power Electron. Motion Control PEMC’90, Budapest, Hungary, Oct.1–3 1990, pp. 83–87. [8] S. Valtchev, J. B. Klaassens, and M. P. N. van Wesenbeeck, “Superresonant converter with switched resonant inductor with PFM-PWM control,” IEEE Trans. Power Electron., vol. 10, no. 6, pp. 760–765, 1995. [9] J. F. Lazar and R. Martinelli, “Steady-state analysis of the LLC series resonant converter,” in Proc. IEEE APEC 2001, vol. 2, pp. 728–735. [10] D. Meeker, “Finite element method magnet,” ver. 33, Users Manual 2003. [11] “MagNet 2D & 3D Electromagnetic Field Simulation Software (User’s Guide),” ver. 6.15, Infolytica Corp., 2004, Montreal, Canada. [12] S. Valtchev, K. Brandisky, B. Borges, and J. B. Klaassens, “Efficient resonant inductive coupling energy transfer using new magnetic and design criteria,” in Proc. IEEE PESC’05, Recife, Brazil, June 2005, pp. 1293–1298.
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Stanimir Valtchev (M’93-SM’08) was born in Lovetch, Bulgaria in 1951. He received the B.S. and M.S. degrees from the Technical University (TU) of Sofia (best of the year), Sofia, Bulgaria, and the Doctor’s degree from the Instituto Superior Tecnico (IST), Lisbon, Portugal, all in electrical engineering. He was with the Institute for Medical Equipment, Sofia, until 1977, when he was admitted as a Researcher at the TU (then VMEI-Sofia), initially in the Industrial Electronics Laboratory, and then in the Manipulators and Robots Laboratory. He was also an Assistant Director of the Centre of Robotics, TU. After 1980, he worked on high-frequency resonant power converters. During 1987 and from 1991 to 1992, he was with the Laboratory for Power Electronics, the Delft University of Technology, the Netherlands, where he was appointed Assistant Professor in 1987. He was the Deputy Dean, responsible for the international students at TU Sofia, during 1990–1994. In 1994, he was invited in Portugal to lead a project on a new soft-transition power converter. He has taught various subjects in different universities and has consulted various institutions in Portugal and in the Netherlands. Since 1988, he has been an Assistant Professor at TU Sofia, and has taught several courses on power supply equipment and power transistor converters to graduate and post-graduate students. He is currently Auxiliary Professor in the Universidade Nova de Lisboa, Portugal. He has published in numerous conferences and journals. His interests include the power converters (esp. high-frequency, soft-switching and resonant), power harvesting, contactless energy transfer, etc. Dr. Valtchev was the recipient of the IEEE Meritorious Paper Award in 1997.
Beatriz Borges (M’82-SM’08) was born in Cascais, Portugal, in1953. She graduated in 1977 and received the PhD in electrical engineering and computer science from Instituto Superior Técnico, Technical University of Lisbon, Portugal, in 1990. She is currently an Associate Professor of power electronics at the Instituto Superior Técnico. She develops research work at the Instituto de Telecomunicações in high frequency hard and soft switching converters, resonant converters, active power filters and power factor correctors, switching audio amplifiers and in EMI problems. Prof. Borges has been the Chair of the IEEE Portugal Chapter of the joint societies Power Electronics, Industrial Electronics and Industry Applications, since 2007.
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Kostadin Brandisky (M’06) was born in Plovdiv, Bulgaria, in 1950. He received the Dipl. Eng. degree in 1973 and the Ph.D. degree in 1983, both in electrical engineering, from the Technical University of Sofia, Bulgaria. He is currently an Associate Professor at the Department of Theoretical Electrical Engineering of the Technical University of Sofia. From 1978 to 1988, he was an Assistant Professor, and in 1989, he became an Associate Professor of electric circuits and electromagnetics at the Technical University of Sofia. From 1992 to 1994, he worked as Postdoctoral Research Fellow at the Laboratory of Electrical Machines and Drives, Katholieke Universiteit Leuven, Leuven, Belgium. His research interests are in numerical methods for analysis of electromagnetic fields, inverse problems in electrical impedance tomography (EIT) and eddy current nondestructive testing, optimization problems, parallel algorithms and CAD systems in electromagnetics. He has published over 70 technical papers and has co-authored 15 books and manuals.
J. Ben Klaassens was born in Assen, the Netherlands, in 1942. He received the B.S., M.S. and Ph.D. degrees in electrical engineering from the Delft University of Technology, the Netherlands. He is currently an Associate Professor at the Control Laboratory, the Delft University of Technology. His work has been concerned with inverter circuits, pulsewidth modulation, and the control of electrical machinery. His research work and professional publications are in the area of converter systems with high internal pulse frequencies for sub-megawatt power levels employing thyristors, power transistors and IGBTs. He has published a variety of papers on series-resonant converters for low and high power applications. He has designed and built prototypes of the early dc-dc to ac-ac series-resonant converters for a wide variety of applications such as electric motors and generators, communication power supplies, radar signal generators, arc welders and space applications. His research interest has been in the area of modelling and control of converters and electrical drives. His current research interests are in mechatronics and robotics. He is a member of the SmartCars research project, where the vehicles are developing into platforms for advanced electronic systems, and where mechatronics is therefore a key technology. New approaches for load-sensing by measuring forces and moments directly are applied for the control of these platforms.
