Maximum Power Transfer versus Efficiency in Mid-Range Wireless Power Transfer Systems

The condition for maximum power transfer of 2-coi ls wireless power transfer (WPT) system is derived fro m circuit analysis and discussed together with the respective WPT system efficiency (η). In the sequence, it is shown that a 4-coils WPT system (which can be divided in source, two communi cation and load circuits) without power losses at the two communication circuits (ideal 4-coils WPT system) presents, from maximum power transfer and efficiency point of view, a performance similar to those of a 2-coils WPT system. The exception is the influ ence of coupling coefficient (k): in 2-coils system η increases as k approaches one, while in ideal 4-coils WPT system η increases as k between the two communication coils approaches zero. In addition, r ealistic 4-coils WPT systems (with power losses at the two communica tion circuits) are also analyzed showing, for instance, that η presents a maximum as a function of k of the communication coils. In order to validate the presented theory, 4 coils were built, and a set up to perform 2coils and 4-coils WPT systems has been carried out. Practical results show good agreement with the developed theo ry.

Fig. 1 shows the equivalent circuit of a 2-coils WPT system.Considering both circuits tuned at the same resonance angular frequency ( = = ), it can be written and where, M12 is the mutual inductance, R1 the total transmitting circuit resistance (including the internal resistances of the source and those of the involved capacitance (C1) and inductance (L1)), and R2 the total receiving circuit resistance (the sum of internal resistances of the involved capacitance (C2) and inductance (L2) -r2-with the load resistance (RL)).
Electric power is calculated multiplying the resistance by the square of the current amplitude so that using (1) and (2) it can be written = and where, P1 and P2 are the electric power dissipated at R1 and R2, respectively.
Taking the derivative of (4) with respect to M12 and making the result equal to zero, after manipulation, yields = This is the MPT condition for a 2-coils WPT system.(That ( 5) is a condition of maximum can be demonstrated making the second derivative of (4) with respect to M12 equal to zero.)Moreover, using 3) and ( 4) it leads to = = .(6) as classical MPT theorem teaches.
For comparison purposes it is interesting to compute the relative power transferred to R2 dividing (4) by ( 6) which gives Dividing the power transferred to R2 (P2) by the total power (P1+P2), the transmission efficiency (%) can be calculate yielding Note that, as also the classical MPT theorem teaches, using ( 5) in (8) gives % =1/2.

B. 4-Coils Circuit
Figure 2 shows the equivalent circuit of a 4-coils WPT system.
where; M12, M23 and M34 are the mutual inductances, and R1, R2, R3 and R4 the total individual circuits resistances.Note that R4 is, in fact, the sum of internal resistances of the involved capacitance (C4) and inductance (L4) -r4 -with the load resistance (RL).
Taking the derivative of (18), also using 2-coils WPT system as a guide, with respect to k23 (M23 would be more general [8] but in 4-coils WPT system L2 and L3 appears in M12 and M34 and in (18) they had been simplified) and making the result equal to zero yields This is the MPT condition for 4-coils WPT system without power losses at the communication circuits.Note that using (19) in ( 17) and ( 18) gives as classical MPT theorem teaches.
The system efficiency (%), also with R2=R3=0, can be defined as Observe that, in case of R2=R3=0, (23) and ( 24) become ( 21) and ( 22), respectively.Moreover, it can be seen that ( 23) and ( 24), due to their format, present a maximum.Thus, taking the derivative of them with respect to k23 (as mentioned before, M23 would be more general [7] but in 4-coils WPT system L2 and L3 appears also in M12 and M34) and making the results equal to zero yields Observe that (25) and (26) show that the points of maximum of (23) and ( 24) are not coincident.In addition, note that in case of R2=R3=0 (25) becomes (19), and (26) does not present a meaning anymore, i.e., η does not have (in sense of derivative zero) a point of maximum.Finally, substituting (25) in ( 23) and ( 26) in (24) it can be seen that the maximum in (23) and ( 24) is always, as expected, less than one.
III. EXPERIMENTAL VALIDATION For practical evaluation of the analysis presented in the previous section, a set of 4 coils, with equal dimensions and shape, was constructed.The coils are circular with diameter of 150 mm and 20 mm of length, wound with 22 turns of enameled copper 19 AWG wire in a single layer way.The coils selfinductances have a similar measured value of 127 µH in the range of 10 to 800 kHz.All measurements of inductances, capacitances, resistances, and resonance frequencies were obtained using an Agilent precision vector impedance analyzer (4294A).
Since the presented analysis has a strong dependence on the coupling coefficient k, at first, considering coaxial coils, the practical behavior of k in function of distance was determined.For this, 102 it has been used a Tektronix signal generator CFG253 to excite one coil, as primary, at a low frequency of 10 kHz, to reduce the influence of the coils´ stray capacitances.Using an Agilent digital oscilloscope MSO6034, the voltages were measured in the primary (v1) and in the open terminals of the secondary coil (v2), while the distance between the coils was varied.Since the current in secondary coil is zero, v1 = .: :; ⁄ and v2 = .: :; ⁄ .As the inductances are equal (L1=L2=L), then 1 = / , after little manipulation it gives 1 = ⁄ .Fig. 3 shows the measured coupling coefficient in a range of 2 to 32 cm (the distances are considered between the closer first turns).To achieve the same resonant frequency at four coils, precision capacitors of 560 pF were used.In fact, in a set of 20 capacitors, 4 were selected by measuring their capacitances (all around 575 pF).It is important to note that even using precision capacitors, each capacitor was chosen specifically for each coil´s inductance, since the inductances also have small variations.In this way, it was possible to select the resonant frequency of 589 kHz with a precision of 100 Hz for the 4 coils.After tuning, the total internal series resistances RS of the LC circuit, at the resonant frequency of 589 kHz, were measured, resulting in similar values of 3.3 Ω.
To measure the relative power transfer and efficiency, it is necessary to know the currents in coil 1 and in the load.In this way, a shunt resistor (R0) of 10 Ω was used in series with coil 1 in all the measurements (the measured resistance value was 9.85 Ω at 589 kHz).Three different loads (Rl) values were used 10, 50, and 100 Ω (exact values of 9.85, 49.7 and 100.24Ω).All resistor presented measured stray inductances of 40 nH at 589 kHz.
A voltage signal (vG) of 7.7 VRMS with 589 kHz was applied on coil 1.To confirm that the resonant frequency has not been changed by external influences of the setup, such as cable capacitances and others, the resonant frequency was confirmed by measuring a minimum voltage point over the RLC 105 Fig. 7. Relative power (P2/P2MAX) as function of distance between coils (for 2-coils set).Dots are measured values for loads of 10, 50, and 100 Ω (R2 total of 13.15, 53, and 103.54 Ω, respectively).Solid curves are calculated using (7).
However, the resistances R2 and R3 are not external resistors, so P2 and P3 could not be measured directly.Thus, considering that all the power, delivered by the generator PG, is dissipated by ohmic losses, the efficiency can be alternatively calculated by PL/PG, where PG = vG⋅i1.Theoretical values are calculated using (23) and (24).
Several different coils arrangements can be conducted with 4-coils WPT systems.Here two approaches are presented.First, it was imposed the same distance between adjacent coils in a range of 2 to 32 cm between them, yielding k12 = k23 = k34.The results for the efficiency and relative power transferred for a load of 100 Ω are given in Fig. 8. Maximum power was measured at distance of 3 cm, whereas the calculated distance is 5 cm.
It can be noted that for distances smaller than approximately 5 cm the inter-couplings between non-adjacent coils (k13, k14, and k24 different from zero) introduce errors, so that theoretical and practical results are not equal to each other.This is in agreement with the theory presented since the influence of non-adjacent coils (k13, k14, k24) are not considered in the derived equations.

IV. CONCLUSION
The MPT conditions of 2-coils and 4-coils WPT system have been derived from circuit analysis and discussed together with the respective system power transfer efficiency, demonstrating that 4-coils system with R2=R3=0 presents, from maximum power transfer and efficiency point of view, a performance similar to those of a 2-coils system.The exception is the influence of coupling coefficient: in 2-coils system η increases as k12 approaches one, while in 4-coils system η increases as k23 approaches zero.Obviously, the condition R2=R3=0 is not attainable in common practical circuits, being used in this work only as a theoretical guide to allow the comparison between 2-coils and 4coils WPT.
In fact, usually a 4-coils WPT system presents power losses at R2 and R3.For this condition, it has been also demonstrated that 4-coils WPT system has, as expected, its ability to transfer power to the load and its efficiency reduced as power losses at R2 and R3 increase.Here it is important to emphasize that, from efficiency point of view, a 2-coils or a 4-coils WPT system with R2=R3=0 are different from a 4-coils WPT system with power losses at R2 and R3 since the later presents a maximum efficiency as a function of k23.
Note that in a series circuit classical MPT theorem teaches that if RSOURCE=RLOAD, power transferred to the load is maximum and is ½, and that, keeping RSOURCE constant, η increases as RLOAD increases.
So that, for a given source (RSOURCE constant), the circuit designer makes RLOAD> RSOURCE if his or her

Fig. 3 .
Fig. 3. Measured coupling coefficient (k) as function of distance between two adjacent coaxial coils.Four coils with same features: 150 mm of diameter, 20 mm of length, 22 turns of copper wire with diameter 0.9 mm.