Network Representations for Wireless Power Transfer Realized with Resonant Inductive Coils Mauro Mongiardo†
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Abstract — Electromagnetic wireless resonant energy links (WREL) can be achieved by using resonant coils coupled via their magnetic fields; these resonant coils must be, in addition, properly coupled to the source and load. By using a simple network modelling we introduce a methodology that allows the simulation of complex structures without the need of fullwave electromagnetic simulations.
L0
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Marco Dionigi∗
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INTRODUCTION
Recently, wireless energy transfer, also referred to with the name of wireless electricity (witricity) or Wireless Power Transfer (WPT), has been considered in a series of papers [1, 2, 3] which have added new developments to the work of Tesla [4]. The latter work was based on the principle that two resonant circuits, resonating at the same frequency, even when weakly coupled, are able to exchange energy. This possibility for medium–range wireless power transfer is finding several areas of applications. In [5] it has been proposed for wireless powering of single-chip systems, while in [6] it has been applied in biomedical engineering for implanted and worn devices. Applications have also been proposed for Electrical Vehicles (EV) [8], thus avoiding the need of cords. Other applications have been suggested in robotics, for sensor networks [7], and even as a backpack which is able to recharge the devices there placed. In [9, 10, 11] we have noted that this type of subject is treated in a very efficient way by using appropriate network representations instead of using the coupled mode theory as originally proposed in [1, 2, 3]. In particular, network representations can play a fundamental role concerning the modeling of systems with multiple transmitters and receivers. The fact of using multiple transmitters can significantly extend the range for wireless power transfer. The other important aspect concerns the modeling of resonators: a full-wave modeling of the resonators can be performed and results of relevant simulations can be represented in networks models which can then be used in order to design and ∗ DIEI, University of Perugia, via G. Duranti 92 06125 Perugia, Italy, e-mail:
[email protected], tel.: +39 0755853653, fax: +39 0755853653 † DIEI, University of Perugia, via G. Duranti 92 06125 Perugia, Italy, e-mail:
[email protected], tel.: +39 0755853653, fax: +39 0755853653.
RL
Figure 1: Basic circuit layout of a two coil wireless power system. analyze more complex systems. 2
Modeling of wireless power transfer systems
We investigate in this section the network representations and analysis methods for WPT systems by considering the simple two loop WPT system with inductive input and output coupling and the multiple coils system. 2.1
Two coils systems
We have considered the setup sketched in figure 1, where the input generator excites the coil L0 and the output power is drawn from coil L3 . The coils L1 , L2 and the capacitance C1 C2 constitute the resonators, while the capacitance and inductance values are chosen in order to provide the same resonant frequency. The resonators and the input and output inductance are arranged at different distances, D1 , D2 , D12 in order to show at the output load RL the maximum power transfer. The aim of the design is to find the coupling resonators dimensions, quality factor and the coils distances (D1 D2 ) that optimize the power transfer of the system for a given D12 . The structure of figure 1 can be easily modeled by an equivalent circuit introducing the mutual inductance between the coils as shown in figure 2.
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Inverter
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Figure 2: Equivalent circuit of a two coil wireless Figure 3: A) Inductive coupling and Impedance inpower system. verter modeling, B) Impedance inverter network, C) equivalent circuit exploiting the impedance inThe well known parameter k, or inductive cou- verters, D) Equivalent circuit network representapling coefficient, depends on the mutual inductance tion. between the two coils as follows: Lab kab = √ La Lb
I1 1V I2 0 A· I3 = 0 0 I4
(1)
where Lab is the mutual inductance between the coils a and b, while La and Lb are their inductance values. Neglecting the inductor coupling L0,3 a further equivalent circuit is obtained as shown in figure 3 exploiting the impedance inverters as coupling device. Considering the circuit in figure 2D we have a network representation of a two coil system. The solution can be easily obtained by considering an appropriate excitation of the network as shown in section 3. Solving the network by the loop currents method, we obtain the following solution matrix A :
(5)
The output voltage across the load is given by: Vout = I4 RL 2.2
(6)
Multiple coils
More complex WPT systems can adopt multiple resonators in order to extend the service area of the device. Moreover, the multiple coils setup can establish a magnetic field with designed uniformity and intensity in a defined region of space. The analysis of a multicoil system can be obtained extending equations (2-6). In the general case no assumption Zin −jωL0,1 0 0 can be made on the mutual couplings of the differ −jωL0,1 Z1 −jωL12 0 A= ent coils. This affects the solution matrix producing 0 −jωL12 Z2 −jωL2,3 0 0 −jωL22,3 Zout non zero elements above and below the second di(2) agonal row. Discarding the capacitive coupling the where the input and output loop impedances are: solution matrix elements can be written as follows: (
Zin = jωL0 + Rs Zout = jωL3 + RL and the resonator impedances are i h 1 + R1 Z1 = j ωL1 − ωC 1i h 1 Z2 = j ωL2 − ωC2 + R2 .
Ai,k =
(3)
−jωLi,ki h 1 j ωLi − ωC + Ri . i
if i 6= k if i = k
(7)
Once, for a certain topology, the matrix A has been computed, the solution is easily obtained by extending equations (5) and (6). (4) 3
SCATTERING PARAMETERS EVALUATION
finally, given the input exciting voltage we obtain To obtain the scattering parameters, by using simthe loop currents as follows: ulators working either in time or in frequency do-
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Mutual inductance calculation
A key feature of the modeling is the computation of mutual inductance between coils inductors. A vast literature has been produced on this topic since the beginning of 1900 [12, 14, 13]. For sake of simplicity we will refer only to single loop coupling. The inductance Lloop of a loop inductor of radius RLoop Figure 4: Circuit schematic for the s parameter cal- and wire diameter R can be modeled as follows w culation by network excitation [12]: LLoop = µ(2Rloop − Rw )
1−
k2 2
K(k) − E(k) .
mains, it is convenient to adopt the circuit depicted (18) in Fig. 4.By slightly modifying the input section, S parameters can be derived by using common cir- where K and E are are complete elliptic integrals cuital solvers (e.g. spice). With reference to Fig. 4, of the first and second kinds and k is given by: being Z0 the reference impedance, we obtain: 4Rloop (Rloop − Rw ) . (19) k= Z1 − Z0 (2Rloop − Rw )2 , (8) S11 = Z1 + Z0 The mutual inductance between two coaxial e loops at distance d and with radii R1 and R2 can Z1 . (9) be computed as follows : V1 = 2 Z1 + Z0 Solving the equation (9) for Z1 : p β2 2 L1,2 = 2µ R1 R2 1− K(β) − E(β) β 2 V1 Z0 Z1 = , (10) (20) 2 − V1 where: and replacing in (8), we obtain: s 4R1 R2 S11 = V1 − 1 (11) β= (21) (R1 + R2 )2 + d2 observing the circuit illustrated in Fig. 4 it is clear that the voltage across A and B represents the S11 5 Results parameter. From equations (18,20) it is possible to analyze a S21 is derived from: wide variety of WPT configurations. A standard V2 − I2 Z0 . (12) configuration, as shown in figure 1, has been manS21 = V1 + I1 Z0 ufactured and measured. The results are shown in figure 5. When the port 2 is matched, we can write: The measurement and the simulated response are V2 in good agreement demonstrating the high accuracy (13) I2 = − , of the analytical network model of the structure. Z0 replacing in (12):
6
S21 = 2
V2 V1 + I1 Z0
Considering that: I1 =
V1 Z1
we obtain for S21 : S21 = 2
1 V2 V1 1 + Z0 /Z1
(14) We have presented a network model of WPT system and the simulation procedure for a complete system. The measured results confirms the good accuracy of the model. The high efficiency attain(15) able for this arrangement where the distance between coils is equal to their diameter makes WPT an excellent method for EV applications. (16)
References
and, replacing (10) in (15), we obtain: S21 = V2 .
Conclusions
(17)
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[1] J.D. Joannopoulos, A. Karalis, M. Soljacic, “Wireless non-radiative energy transfer," US Patent Application Publication, Pub. No. US2007/0222542, pub. date sep. 27, 2007.
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5 0 -5 -10
|S11| Measured |S21| Measured |S11| Simulated |S21| Simulated
Amplitude (dB)
-15 -20 -25
[7] M.K. Watfa, H. Al-Hassanieh, S. Salmen, “The Road to Immortal Sensor Nodes," ISSNIP 2008. International Conference on Intelligent Sensors, Sensor Networks and Information Processing,, Sydney, NSW, 2008, Pages 523 - 528.
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Measured Simulated
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[10] M. Dionigi, M. Mongiardo, “CAD of wireless resonant energy links (WREL) realized by Frequency (MHz) coils," MTT-S International Microwave Symposium, Anaheim, CA , USA, pp. 1760 - 1763, 2010. Figure 5: WPT system measured and simulated scattering parameters and efficiency . R1 = R2 = 5 [11] M. Dionigi, P. Mezzanotte, M. Mongiardo, cm, D12 = 5 cm, D1 = D2 = 1.3cm and C1 = C2 = “Computational Modeling of RF Wireless Res19.46 pF. onant Energy Links (WREL) Coils–based Systems," Aces, Tampere, Finland, April 25–29, 2010. [2] A. Karalis, J.D. Joannopoulos, M. Soljacic, “Efficient wireless non-radiative mid-range energy transfer," Annals of Physics, Elsevier, [12] I. Bahl, Lumped Elements for RF and Microwave Circuits ,Artech House, 2003 323, pp. 24-48, 2008. 0
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