Series Resonant Converter Applied to Contactless Energy Transmission Stanimir Valtchev *, Beatriz V. Borges+ and J.B. Klaassens! Instituto de Telecomunicações *+, Instituto Superior Técnico + Av. Rovisco Pais 1046-001 Lisboa Codex, PORTUGAL Tel. 351 218418379; Fax. 351 218418472; Email:
[email protected] !
Delft University of Technology Mekelveg 4, 2628CD, THE NETHERLANDS Tel. 31 15 2782928; Email:
[email protected]
Abstract This paper presents the theoretical analysis, simulation and design optimisation of a Series Resonant Converter (SRC) working as an interface for inductive coupling. Inductive coupling is needed when the transfer of energy has to be made without any electrical contact. In this case the coupling transformer has to be separable, i.e., the transformer parameters are time varying variables, instead of constants. It is shown that, the operating region that leads to the highest efficiency can be identified. By appropriate control the SRC is insensitive to the transformer parameters’ variations and therefore, it is one of the most suitable topologies to be used in contactless energy transmission. I. INTRODUCTION During the decades of 80-s and 90-s was witnessed a great effort of investigation in the area of resonant converters, having as one of the principal aims, the increase of the switching frequency in order to obtain converters with higher power density, i.e. small dimensions and high efficiency. Resonant converters operate pushing the semiconductor devices to turn on and/or turn off at the instants of extinction of current or voltage, in order to reduce the switching losses, facilitating the increase of the frequency without the corresponding increase of stress and degrading of the efficiency. As a result of this development, converters were achieved with much better characteristics in terms of switching frequency, bandwidth and efficiency. However the only observable adhesion of the industry to this principle of power conversion, was only in several special cases. The industry has not adopted immediately the large-scale production of this class of converters due to the following reasons: (1) - the existence of two reactive loops of dissimilar dynamics (the resonant loop and the output filter) provokes an increased complexity in converter dimensioning. It is not possible to predict the waveforms of the electrical variables as easy as in case of hardswitched converters (which technology is well known in the industry); (2) - the control strategies definition and in the inherent stability study of the systems is more complex; (3) - although it is possible to annulate, in theory, the commutation losses in converters based on resonant technology, in some applications the peak values of the
electrical variables may reach excessive levels, provoking higher conduction losses and oversising of the converter components. Although at the industry level the use of resonant technology has not been widespread yet, the converters using this technology, show so great capabilities that, once minimised the drawbacks referred to, rate these converters as the best or even the only solution for specific applications, either for low power level or for medium/high power level. The use of resonant converters may be not only extremely advantageous but also indispensable for the on-going development of mobile robotics, on-board power systems in electrical or hybrid vehicles or in space telecommunications. Arises for example, the problem to discover and analyse converter configurations that allow optimisation and stabilisation of the energy transfer with the best possible efficiency, in applications where the isolation transformer is not stationary as in conventional power supplies, but whose primary and secondary are in continuous movement relative to each other and demonstrating a variable mutual inductance coefficient. This problem has no simple solution if conventional hard-switching power converters are used, because for example, the leakage inductance could result in destructive effects on the vulnerable power semiconductor elements. In contrary, the resonant converters because of their principle of operation, could easily adapt themselves to the variation of the reactive components in the resonant loop by varying the operating frequency or other control parameters. The investigation in the field of loosely coupled transformers is still in the beginning, showing up only a few recent articles [1-3] focussing on the problems of modelling magnetic components for application in contactless energy transfer. The objective of this paper is to present a first approach to the theoretical analysis, simulation and design optimisation of a Series Resonant Converter working as an interface for inductive coupling. The study presented is based on a previous work, [4] where the SRC was analysed when using an ideal transformer. The first section of the paper is dedicated to the review of the principles of operation of the SRC and to its analysis. Characteristic curves representing the variation of the steady state voltage gain q with the normalized output current and with the normalized switching frequency are presented. Considering the current form factor, the definition
of the operating region where the highest efficiency is obtained is also defined. The second section of paper will be concerned with the transformer modelling, where the primary and secondary leakage inductances and the magnetizing inductance are variables. Some considerations related to the operation of the with such a transformer are then established. Simulation and experimental results are presented in order to verify the theory. II. REVISION OF SRC OPERATION A . Operating principle Fig.1 shows the schematics of the power circuit of the Series Resonant Converter. The circuit operation consists in closing the pairs of switches Q1,Q2 and Q3,Q4 alternatively at a frequency above the resonant frequency of the resonant circuit composed by Lr and Cr. Q3
uC
Lr
+
-
ES
n:1
CO
iL uAB Cr Q4
Q2
SRC operating modes Mode
I
II
III
IV
uAB
+ES
- ES
- ES
+ ES
uT
nUO
nUO
- nUO
-nUO
ULC
ES -nUO
-ES -nUO
-ES +nUO
ES +nUO
Switch on
Q1 ,Q2, DA
D3, D4,DA
Q3,Q4, DB
D1,D2, DB
B . Circuit Analysis: Considering the equivalent circuit of Fig.2 and using a normalized notation where all the voltages are divided by the input voltage, ES, and currents multiplied by Z/ES the state matrix equation of the series resonant converter is d ZiLN (ωt ) 0 − 1 ZiLN (ωt ) U LN = + dt uCN (ωt ) 1 0 uCN (ωt ) 0
DA
+ A
B
-
Q1
Table I
uT
+ UO -
DB Fig.1 Electrical circuit of the series resonant DC-DC converter.
Operation may be obtained at a switching frequency below or above the resonant frequency, however it is desirable to operate above the resonant frequency in order to reduce the switching losses [4]. Operation above the resonant frequency produces an inductive behaviour of the circuit, generating a lagging alternating current iL, in the resonant circuit. Providing that the output capacitor CO is sufficient to consider that the output voltage does not change significantly, during a few periods of operation, the action of the rectifier diodes DA and DB impose at the primary terminals of the power transformer, an alternating square wave voltage uT, synchronous with the resonant current, with an amplitude of nUO. The SRC is, therefore, equivalent to an LrCr circuit supplied by two alternating square voltages uAB (generated by the action of the switches Q1,Q2, Q3 and Q4) with an amplitude equal to the input voltage, and voltage uT (generated by the action of the output diodes DA and DB), as presented in Fig. 2.
(1)
where Z is the characteristic impedance defined by Lr and Cr and Zi LN (ωt ) = ZiL (ωt ) / E S is the normalized resonant current, uCN (ωt ) = u C (ωt ) / E S is the normalized resonant voltage, and N U LC = U LC / E S is the normalized excitation voltage. uGQ1 uGQ2 t
uGQ3 uGQ4
t
ES uAB
t
- ES
t
nUO uT - nUO
mod I
II mod III
IV
t
iL t
iL uAB
+ -
Lr
+
Cr uC
+
uT
-
uC
t
Fig.2. Series Resonant DC-DC Converter equivalent circuit.
The converter has four modes of operation defined by the four combinations of the values of the two power supplies. Table I identifies the operation modes and the correspondent conducting switches and generated voltages.
Fig. 3 Series resonant DC-DC converter: steady state variable waveforms.
Integrating (1) the general solution is obtained: N N uCN (ωt ) = U LC − [U LC − uCN (0) ]cos ωt + Zi LN (0) sin ωt
[
]
N Zi LN (ωt ) = U LC − u CN (0) sin ωt + Zi LN (0) cosωt
(2) (3)
As described in Table I, each mode corresponds to a different excitation voltage ULC. However, the values of ULC corresponding to modes I and III, and to modes II and IV are symmetrical, respectively, and therefore, the analysis can be performed by considering only the positive arcade of the resonant current i.e., considering mode I, corresponding to the time interval [0,xk] and mode II, corresponding to time the interval [xk,x0] (Fig.3). Substituting the initial conditions:
iLNmod I (0) = iLNmod II ( x0 ) = 0
u
(0) = −u
N C mod I
N C mod II
i
( xk ) = i
N L mod II
( xk )
(5)
]
N N N uCN (ωt ) = U LC mod I − U LC mod I − uC (0) cos ωt
]
N N ZiLN (ωt ) = U LC mod I − uC (0) sin ωt
0
1
2
3
4
Fig.4 Normalised output voltage gain q in function of normalized switching frequency for different values of the current form factor ρi.
q ρ =1.08 ρ=1.09 ρ=1.1 ρ=1.11 ρ=1.12 ρ=1.13 ρ=1.14
(6) (7)
and for Mode II:
[
]
N N N uCN (ωt ) = U LC mod II − U LC mod II − uC ( x0 ) cos(ωt − x0 ) (8)
[
Zi (ωt ) = U N L
N
with U LC mod I
N LC mod II
]
N = 1 − q ,U LC mod II = −1 − q
Considering that the energy in the LrCr tank calculated over the time interval (0,x0) is null, and attending to equations (6) to (9) the time intervals xk and x0-xk can be obtained [4]:
1 + q + quCNmax x0 − xk = arccos N 1 + q + uC max with
u CNmax
equal to
u CN ( x 0 )
or to
(10)
2
3
4
5
ION
Fig.5 Normalized output voltage gain q as a function of the normalized output current for different values of the current form factor ρi.
Inspection of Fig. 4 indicates that the best performance in efficiency is obtained for higher values of q and for a switching frequency above, but close to the resonant frequency. 1 FN=1.05 q FN=1.1 FN=1.2
(11) FN=1.5
− u CN (0) .
III. CURRENT FORM FACTOR AND EFFICIENCY In [4] is shown that the efficiency achieved in the SRC inversely varies with the square of the resonant current form factor, defined by ρ = I rms / I O :
i
R 1 2 = 1 + loss ρ i η Ro
1
(9)
,and q=nEO/ES.
1 − q − quCNmax xk = arccos N 1 − q + uC max
0 0
− u ( x0 ) sin (ωt − x0 ) N C
5F N=fS/fres
1
in equations (2) and (3), the state variables equations are obtained for Mode I:
[
ρ =1.08 ρ=1.09 ρ=1.1 ρ=1.11 ρ=1.12 ρ=1.13 ρ=1.14 ρ=1.15
( x0 )
uCNmod I ( xk ) = uCNmod II ( xk )
[
1 q
(4)
and the boundary conditions at time xk (xk is the time instant where the transition between modes I and II takes place, Fig.3) N L mod I
where Rloss is the equivalent losses resistance of the circuit, and Ro is the load resistance. As expressed in (12) the highest efficiency is obtained for smaller values of ρi. Considering the previous analysis, characteristic curves showing the behaviour of the SRC for constant form factor are obtained and presented in Figs. 4 and 5.
(12),
FN=2 F =5 0 0 1 N
2 3 4 5 IO N Fig.6 Normalized output voltage gain q as a function of normalized output current for different values of the normalized frequency.
On the other hand, Fig.5 shows that higher efficiency is obtained for higher values of q and lower values of the output current. By intersecting the curves in Figs. 4 and 5, the parameter ρi can be eliminated and a new set of characteristics representing the variation of the conversion ratio q in function of the normalized load current are obtained and presented in Fig.6. The above conclusions indicate that the optimal efficiency is obtained for the working points inside the region marked by the dotted line in Fig.6. This theoretical result is in agreement with the practical results obtained in a prototype. IV. TRANSFORMER MODELING AND CURRENT DISTRIBUTION. For contactless energy transfer the magnetic coupling is not ideal, as was considered in the previous study. In this case the leakage and magnetizing inductances of the power transformer vary accordingly with the distance between the two magnetic parts of the transformer. It is expected that the optimal operating region, defined above (Fig. 6), will deviate, depending on the distance between the two parts. In order to study the behaviour of the SRC in this situation, it is of great importance to model the power transformer. Fig. 7 represents the transformer model used in simulation with n=1. Lr
Cr -
+ uAB
+
Llk1 iL
uC
Llk2 iL2
iLm
+
Lm
impossible, so, the objective is to find a way to minimize its influence on the overall efficiency of the converter. The previous defined operating conditions that lead to the highest efficiency, are not expected to be maintained in this situation. The reason for this is that high values of the normalized output voltage prevent a considerable part of the resonant current from being transferred to the load, (i.e. higher q produces lower iL2). The deviated amount of current is circulating in the resonant circuit, causing only additional losses. In addition, lowering the output voltage, although contributing for a better distribution of the resonant current, decreases the output power considerably, and therefore the efficiency becomes reduced. It is important to define a good compromise between the high output voltage (worse distribution of the current) and the better resonant current distribution (lower output voltage). Theoretical studies with the objective to define this optimum region are being carried out and will be subject of future work. However, some simulations with realistic values that support the above considerations have been accomplished. V. SIMULATIION The converter has been simulated using pSPICE, for different values of the air gap and different values of the conversion ratio q. The following parameters were used in simulation: Input voltage: 800V; Output voltage: 200 – 600V; Resonant frequency=30kHz and switching frequency =42kHz.
uT
-
-
iL2
Fig.7 SRC equivalent circuit including transformer model.
iLm
It has been considered that as the distance between the two transformer parts increases, the magnetizing inductance decreases significantly, while the leakage inductances vary slightly: the primary decreases and the secondary increases. Table 2 indicates the values used in simulation. Table II Variations of transformer magnetic parameters with air gap. Air gap [mm] ∆=0 ∆=0.5 ∆=1
Lm [mH]
Llk1 [µH]
Llk2 [µH]
1438 427 297
26.5 17.6 16
108,2 124.2 128.0
Considering the equivalent circuit of Fig. 2 where an ideal transformer is used, the totality of the resonant current is transferred to the load. This does not happen in the equivalent circuit of Fig.7. Because the magnetic coupling is not ideal, the resonant current will be divided into two currents: iL2 and iLm. Only part of the resonant current, iL2, contributes to the load current. This fact produces additional losses in the circuit that are inherent to contactless power transfer, the magnetizing current iLm. will circulate with no contribution to the load current. The complete elimination of this loss is
Fig. 8 Magnetizing current iLm and current iL2 for q=.75 and ∆=0.01mm.
iLm iL2
Fig. 9 Magnetizing current iLm and current iL2 for q=.75 and ∆=1mm.
The simulation results presented in Fig.8, 9 and 10 show the magnetizing current and current iL2 for different values of the air-gap and the output voltage. This results are in accordance to the previous considerations: for higher values of output voltage and worse magnetic coupling the value of the magnetizing current is near the double of the to be transferred to the output (Figs. 8 and 9). To compensate this difficult transfer of current to the output, the output voltage must be decreased (for example by using a different transformer ratio). This is illustrated in Fig.10 where iL2 has a more significant value than the magnetizing current for the same bad magnetic coupling that was used for the results in Fig.9.
in order to increase the output current, thus changing q to 0.25.
Voltage in one switch
iL
iL2 iLm
Fig. 12 Experimental results: switch voltage and resonant current (iLm+iL2) for ∆= 0.5mm.
Voltage in one switch
Fig. 10 Magnetizing current iLm and current iL2 for q=.25 and ∆=1mm.
VI. EXPERIMENTAL RESULTS In order to verify the circuit feasibility, preliminary experimental results have been obtained in a lab prototype. VI=100V, f=60kHz, VO=25V (n=3 e q=.75) and different airgaps. Fig.11 corresponds to the normal high efficiency operation.
Voltage in one switch iL
Fig 11 Experimental results: switch voltage and resonant current (iLm+iL2) for ∆= 0.01mm.
As it can be seen in Figs. 12 and 13 the shape of the resonant current is nearly triangular which means that a great part of this current is circulating in the magnetizing inductance without contributing to the output power. This means that a different transformer ratio (n=1) should be used
iL
Fig. 13 Experimental results: switch voltage and resonant current (iLm+iL2) for ∆= 1mm.
VII. CONCLUSIONS A first approach to studies on the SRC operation when applied to contactless energy transfer is presented. This kind of application is extremely adverse to the converter operation. It is shown that by proper choice of the SRC parameters the converter efficiency can be greatly improved. Simulation and experimental results are presented to support the study. VIII. REFERENCES [1] - J. Hirai, T. Kim and A. Kawamura, “Wireless Transmission of Power and Information for Cableless Linear Motor Drive”, IEEE Transactions on Power Electronics, vol.15, n.1 Jan. 2000, pp. 21- 27. [2] - J. Barnard, J. Ferreira and J. van Wyk, “Sliding Transformers for Linear Contactless Power Delivery”, IEEE Transactions on Industrial Electronics, vol. 44 n.6 Dec. 1997, pp 774-779. [3] - D. Pedder, A. Brown and J. Skinner, “ A Contactless Electrical Energy Transmission System”, IEEE Transactions on Industrial Electronics, vol. 46 n.1 Feb.1999, pp 23-30. [4] – S. Valtchev, J. Klassens, “Efficient Ressonant Power Conversion”, IEEE Transations on Industrial Electronics, vol.37, nº6 Dec.1990, pag.490-495.
