ELECTROMAGNETIC SYSTEM DESIGN FOR WIRELESS POWER
By
JOAQUIN JESUS CASANOVA
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA 2010 °c 2010 Joaquin Jesus Casanova
2 To my family, for their support and encouragement
3 ACKNOWLEDGMENTS First and foremost, I’d like to thank Dr. Jenshan Lin for being the most helpful, understanding, and encouraging advisor a student could ask for. He is one of the rare professors who will give his students to explore their research on their own, and in doing so, allows them to truly learn. Thanks are also due to my commitee, Dr. Henry Zmuda, Dr. Robert Moore, and Dr. Subrata Roy, for their encouragement and insightful questions. They put me at ease without going easy on me. I owe a debt of gratitude to Zhen Ning Low for taking the first steps on this project, for his help understanding power amplifiers, and for his friendship and conversation. He kept me sane. I thank Jason Taylor, Ashley Trowell, and Raul Chinga for their technical support, guidance, and friendship in working on this project. Thanks also go out to my parents and my brother, who always supported me, even if it did seem like my life was nothing but my research to the exclusion of all else. Finally, I’d like to thank Florida High Tech Corridor and Florida Department of Environmental Protection for funding and support.
4 TABLE OF CONTENTS page
ACKNOWLEDGMENTS ...... 4
LIST OF TABLES ...... 8
LIST OF FIGURES ...... 9 ABSTRACT ...... 14
CHAPTER 1 INTRODUCTION TO WIRELESS POWER TRANSFER ...... 16
2 LOOSELY-COUPLED NEAR FIELD WIRELESS POWER ...... 18 2.1 Introduction ...... 18 2.2 Analysis ...... 18 2.2.1 Design Equation for Crx ...... 20 2.2.2 Design Equation for Lout ...... 21 2.2.3 Design Equation for Cout ...... 21 2.2.4 Design Equation for Ct ...... 22 2.3 Tests ...... 23 2.4 Conclusion ...... 28 3 NEAR-FIELD ELECTROMAGNETIC ANALYSIS ...... 30
3.1 Introduction ...... 30 3.2 Coil Fields ...... 30 3.3 Coil Inductance ...... 31 3.4 Coil Parasitics ...... 33 3.4.1 Round Conductor ...... 33 3.4.1.1 Skin effect ...... 33 3.4.1.2 Proximity effect ...... 35 3.4.2 Rectangular Conductor ...... 36 3.4.2.1 Skin effect ...... 37 3.4.2.2 Proximity effect ...... 40 3.5 Litz Wire ...... 42 3.6 Regulations ...... 42 3.7 Conclusion ...... 43
4 OPTIMAL PRIMARY COIL DESIGN ...... 45
4.1 Introduction ...... 45 4.2 Planar Wireless Power System ...... 45 4.3 Coil Design ...... 46 4.4 Testing ...... 47
5 4.5 Results ...... 48 4.6 Conclusion ...... 49
5 M:N ANALYSIS ...... 52
5.1 Introduction ...... 52 5.2 Analysis ...... 52 5.3 Tests Results ...... 55 5.3.1 Verification ...... 57 5.3.2 Receiver Decoupling ...... 58 5.3.3 Impact on Efficiency and Total Received Power ...... 63 5.4 Conclusion ...... 65
6 OPTIMAL PRIMARY COIL DESIGN FOR MULTIPLE COILS ...... 67
6.1 Introduction ...... 67 6.2 Coil Design ...... 67 6.3 System ...... 68 6.4 Testing ...... 71 6.5 Results ...... 71 6.6 Conclusion ...... 73 7 INCLUSION OF FERRITES ...... 74
7.1 Introduction ...... 74 7.2 Inductance Estimation ...... 75 7.3 Loss Estimation ...... 78 7.4 Thickness and Width Effects ...... 78 7.5 Experimental Evaluation ...... 80 7.6 Conclusion ...... 83
8 BAYESIAN LOAD/FAULT TRACKING ...... 84
8.1 Introduction ...... 84 8.2 Technology/Data ...... 86 8.3 Theory/Methods ...... 86 8.3.1 State/Measurement Model ...... 86 8.3.2 Particle Filter Algorithm ...... 87 8.3.2.1 Dataset generation ...... 87 8.3.2.2 Initialization ...... 88 8.3.2.3 State ...... 88 8.3.2.4 Measurement ...... 88 8.3.2.5 Update ...... 89 8.3.2.6 Estimate ...... 89 8.3.3 Tests ...... 90 8.3.4 Implementation ...... 90 8.4 Simulation Results ...... 91
6 8.5 Measured Results ...... 95 8.6 Conclusion ...... 96
9 MIDRANGE WIRELESS POWER TRANSFER ...... 103
9.1 Introduction ...... 103 9.2 Analysis ...... 104 9.2.1 Coil Design ...... 105 9.2.2 Component Selection ...... 106 9.2.2.1 Series-parallel ...... 107 9.2.2.2 Series-series ...... 108 9.2.2.3 T-network ...... 108 9.3 Preliminary Tests ...... 109 9.3.1 Rectifying Diode Effects ...... 109 9.3.2 Frequency and Inductance Effects ...... 111 9.3.3 Topology Effects ...... 113 9.3.4 Sensitivity ...... 113 9.4 Synthesis ...... 118 9.4.1 50 cm Separation ...... 118 9.4.2 1 m Separation ...... 119 9.5 Conclusion ...... 123
10 FAR-FIELD WIRELESS POWER TRANSFER ...... 125
10.1 Introduction ...... 125 10.2 Theory ...... 127 10.3 Solution Details ...... 128 10.4 Physical Properties ...... 129 10.4.1 Soil ...... 129 10.4.2 Atmosphere ...... 130 10.4.2.1 Gaseous water vapor ...... 130 10.4.2.2 Water droplets ...... 131 10.4.2.3 Ice crystals ...... 133 10.4.3 Vegetation ...... 134 10.5 Results and Discussion ...... 135 10.5.1 Atmospheric Loss Estimation for Solar Power Satellite ...... 136 10.5.2 Loss Estimation for Radiofrequency-Harvesting Sensor Under Vegetation Canopy ...... 138 10.5.3 Flux ...... 139 10.6 Conclusions ...... 139 11 CONCLUSIONS ...... 142
REFERENCES ...... 143
BIOGRAPHICAL SKETCH ...... 150
7 LIST OF TABLES Table page
2-1 Design parameters...... 23
2-2 Component values...... 23
4-1 Summary of system performance...... 49 5-1 Component values for 1 and 2 transmitter systems...... 57
5-2 Maximum Prx and maximum ηc for different M:N arrangements...... 63 6-1 Design parameters...... 68
6-2 Component values...... 70 6-3 Summary of system performance...... 71
7-1 Ferrite properties...... 81
7-2 Ferrite experimental evaluation with solenoid coil...... 82
9-1 Component values...... 110
9-2 Component values...... 111 9-3 Component values...... 112
9-4 Component values...... 118
9-5 Summary of 1 m tests...... 120
10-1 Parameter values for RTE...... 136
8 LIST OF FIGURES Figure page
2-1 One-to-one wireless power system block diagram...... 18
2-2 Class E driving circuit for a wireless power system...... 19
2-3 Test setup...... 24
4 2-4 From top left, clockwise: Rin, ∠Ztx , drain voltage waveform at RL = 10 Ω, and Q...... 25
2-5 Received power and total efficiency as a function of RL...... 26
2-6 Received power and total efficiency as a function of RL, and their 95% confidence intervals...... 27
2-7 Efficiency as a function of RL...... 28 3-1 Current stick for MQS analysis...... 31 3-2 Magnetic field components using MQS and MoM techniques of a 1m by 1m square coil...... 32
3-3 Magnetic field magnitude using MQS and MoM techniques of a 1m by 1m square coil...... 32
3-4 Conductor cross section showing field and current in round conductor under skin effect...... 34 3-5 Conductor cross section showing field and current in round conductor under proximity effect...... 35
3-6 Conductor cross section showing field and current in rectangular conductor under skin effect...... 37
3-7 Conductor cross section showing field and current in rectangular conductor under proximity effect...... 40
4-1 Transmitter test setup...... 44 4-2 Coil layout...... 45
4-3 Calculated z-directed magnetic field, assuming 1 A current (A/m)...... 46
4-4 Field probe measurement (mV)...... 47
4-5 Received power (W) as a function of the location of the center of the receiving coil...... 49
4-6 Power (W) and efficiency (%) at loads from 10 Ω to 2 kΩ...... 49
9 5-1 M:N block diagram...... 52 5-2 Coil arrangements...... 55
5-3 Measured vs. predicted Prx ...... 57 5-4 Power space plots for two-receiver tests with small receivers...... 58
5-5 Power space plots for two-receiver tests with large receivers...... 58 5-6 Power space plot for three-receiver test...... 60
5-7 Power vs. efficiency plot for two-receiver tests with small receivers...... 60
5-8 Power vs. efficiency plot for two-receiver tests with large receivers...... 61
5-9 Power vs. efficiency plot for three-receiver tests with small receivers...... 61
5-10 Total received power as a function of RL, and its 95% confidence intervals . . 63
5-11 Total efficiency as a function of RL, and its 95% confidence intervals...... 64 6-1 Coil layout...... 68
6-2 Calculated z-directed magnetic field, assuming 1 A current (A/m)...... 68
6-3 Transmitter test setup...... 69 6-4 Overlap of dual transmitter coils...... 69
6-5 Received power (W) as a function of the location of the center of the receiving coil...... 71
6-6 Power (W) and efficiency (%) at loads from 75 Ω to 4 kΩ...... 71 7-1 Diagram of ferrite shielding...... 73
7-2 Empirical µeff predictions (red x) and observations (blue circle)...... 76 7-3 Flux-field hyteresis loop ...... 76
7-4 Effects of thickness and relative width of ferrite on inductance...... 78
7-5 Effects of thickness and relative width of ferrite on resistance...... 79
8-1 Generated measurements in (Vin,IDC ) space...... 90 8-2 True (blue) and estimated (red) mode for N=10, with resampling...... 91
8-3 True (blue) and estimated (red) mode for N=100, with resampling...... 92
8-4 True (blue) and estimated (red) mode for N=1000, with resampling...... 93
10 8-5 True (blue) and estimated (red) mode for N=10, without resampling...... 94 8-6 True (blue) and estimated (red) mode for N=100, without resampling...... 95
8-7 True (blue) and estimated (red) mode for N=1000, without resampling. . . . . 96
8-8 RMSE of mode and states...... 97
8-9 Mode and charge estimate variance, with resampling...... 97 8-10 Mode and charge estimate variance, without resampling...... 98
8-11 Test of different modes in (Vin,IDC ) space, in real system...... 98 8-12 True (blue) and estimated (red) mode for N=100, without resampling, in real system...... 99 8-13 Predicted and observed power, resistance, and input voltage, and DC input current for N=100, without resampling, in real system...... 99
8-14 True (blue) and estimated (red) mode for N=1000, without resampling, in real system...... 100
8-15 Predicted and observed power, resistance, and input voltage, and DC input current for N=1000, without resampling, in real system...... 100 8-16 True (blue) and estimated (red) mode for N=10000, without resampling, in real system...... 101
8-17 Predicted and observed power, resistance, and input voltage, and DC input current for N=1000, without resampling, in real system...... 101
9-1 Midrange class E series-parallel architecture...... 106
9-2 Midrange class E series-series architecture...... 107
9-3 Midrange class E T network architecture...... 108
9-4 Diode effects on system performance...... 109 9-5 Frequency and inductance effects on system performance...... 110
9-6 Topology effects on system performance...... 111
9-7 Effect of D, d, and f on total efficiency at N=8...... 113
9-8 Effect of D, d, and f on received power at N=8...... 113 9-9 Effect of N, f , and D on total efficiency where D = d...... 114
9-10 Effect of N, f , and D on received power where D = d...... 114
11 9-11 Coil offset effects on system performance...... 116 9-12 Efficiency at nominal component values (black line) and 95% confidence intervals.117
9-13 Power at nominal component values (black line) and 95% confidence intervals.118
9-14 50 cm system performance...... 120
9-15 1 m system setup...... 121 9-16 1 m system performance...... 122
10-1 An example of a radiofrequency (RF) harvesting wireless sensor node [3]. . . 124
10-2 An illustration of the Solar Power Satellite (SPS) concept [61]...... 124
10-3 Atmosphere model schematic used in this chapter [63]...... 125
10-4 Canopy model schematic used in this chapter [64]...... 125 10-5 Cloud droplet distribution and absorption cross section...... 131
10-6 Rain droplet distribution and absorption cross section...... 132
10-7 Cloud droplet distribution and scattering cross section...... 133
10-8 Rain droplet distribution and scattering cross section...... 134 10-9 Ice sphere distribution and absorption cross section...... 135
10-10 Ice sphere distribution and scattering cross section...... 136
10-11 Log intensity distribution through cloud...... 137
10-12 Log intensity distribution through rain...... 138
10-13 Log intensity distribution through ice...... 139 10-14 Log intensity distribution through vegetation with diffuse and specular lower boundary...... 140
10-15 Flux profile through different media...... 140
12 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ELECTROMAGNETIC SYSTEM DESIGN FOR WIRELESS POWER
By
Joaquin Jesus Casanova
May 2010
Chair: Jenshan Lin Major: Electrical and Computer Engineering
Wireless communications technology has freed electronics from communication cables. The natural next step is to cut the last wire of portable wireless devices, the power cable. Wireless power systems would permit charging many different devices equipped with receiving coils, in addition to delivering power through rooftops and through the atmosphere. The approaches to wireless power transfer can be categorized as near-field, midrange, and far-field. To date, the latter is still impractical for consumer applications due to the high power and large antenna requirement necessary to achieve levels of power comparable to a wall supply. On the other hand, near-field inductive coupling has more promise as a wireless power technology for charging battery-operated devices. Midrange power transfer has the most potential for applications such as vehicle charging and power transmission through walls and rooftops. Far-field applications include radiofrequency (RF) energy harvesting and transmission of power from space.
This dissertation presents several apsects of the design and testing of wireless power systems. Circuit topology and electromagnetic design of a near-field system is considered, as well as the extension of the system to multiple coils. In addition, the use of ferrite shielding and detection and estimation algorithms are considered for the near-field system. The near-field architecture is extended and modified for a midrange
13 system. Finally, far-field power transfer through the atmosphere and the environment are considered through the numerical solution of the radiative transfer equation.
14 CHAPTER 1 INTRODUCTION TO WIRELESS POWER TRANSFER
The large number of battery operated consumer electronics and the associated tangle of wall-wart chargers has generated interest in designing a single, convenient charging platform [1]. Wireless battery charging systems would permit charging many different devices equipped with receiving coils and cut the last wire of portable wireless devices. Several techniques exist for transmitting power by electromagnetic fields. They differ primarily by the distance between receiver and transmitter (D) and the characteristic dimension of the transmitter (d), relative to wavelength (λ), and by their typical power levels and applications [2]. Far-field, or radiative, power transfer occurs when the distance between the transmitter and the receiver exceeds the Rayleigh distance, D > 2d 2/λ and d > λ. At nearer distances, (D ' d) is considered midrange.
Where D << d electromagnetic coupling is considered near-field. Far-field power transfer is impractical for consumer applications due to the high power and large antenna requirement necessary to achieve levels of power comparable to a wall supply [2]. Two major applications of radiative wireless power transfer (WPT) are in ambient radiofrequency (RF) harvesting and the Solar Power Satellite (SPS). The idea behind the first technique is to convert the radio waves from communications into power [3, 4]. The SPS is an idea which came about in the late 1960s [5], where the principle is to collect solar energy in space using a satellite and beam it to a receiving station on earth.
To date, midrange has shown promise in theory but practical tests at high power levels are lacking in the literature. This is an appropriate range for applications such as wireless charging of electric vehicles [6]. Evanescent couplng employs resonant structures to ensure a strong link between transmitter and receiver [7] in this regime.
On the other hand, near-field inductive coupling has more promise as a consumer-level wireless power technology. This is the familiar prinicple used in transformers and AC
15 and DC machinery. It can also be exploited for WPT to charge battery-operated devices [8–11].
Because of this, this dissertation will focus on the design of a near-field system:
Chapters 2-7 consider the near-field system; Chapters 9-10 consider the midrange and far-field systems, respectively. Chapter 2 discusses a near-field wireless power system circuit architecture and design rules for this circuit. Chapter 3 discusses the electromagnetic theory behind calculation of coil properties of inductance and resistance. A technique for the optimal design of the primary coil is discussed in Chapter 4. The architecture is extended to a system with an arbitrary number of transmitters and receivers in Chapter 5. Coil design for multiple transmitting coils in parallel is discussed in Chapter 6. Chapter 7 discusses the use and evaluation of ferrite shielding. Chapter
8 describes the development and testing of a Bayesian tracking algorithm for receiver discrimination and charge status determination in the near-field system. The extension of the system and coil design to midrange is presented in Chapter 9. Chapter 10 describes the use of radiative transfer modeling for estimating losses of far-field wireless power transmission. Chapter 11 presents some concluding remarks.
16 CHAPTER 2 LOOSELY-COUPLED NEAR FIELD WIRELESS POWER
2.1 Introduction
Fig. 2-1 shows a block diagram for a generalized wireless power system and the
circuit diagram is shown in Fig. 2-2. The inverter is a class E amplifier [12] driven by
a low-power clock at 240 kHz, followed by a series-parallel impedance transformation
network [13]. Selecting the values of Crx , Lout , Cout , and Ct for optimum performance of the wireless power system presents a challenge. [14] shows the design methodolgy for a similar architecture with closed loop control. [15] demonstrates how to choose design values for a class E without relying on Raab’s waveform equations; however, it involves numerical root finding. [16] presents a selection technique for an open-loop system that relies on sweeping component values numerically until the impedance and drain voltage satisfy certain constraints. While this method successfully finds appropriate component values, it is time consuming. This chapter derives simple formulas for the optimum component values by applying the same constraints.
2.2 Analysis
The optimum values for Crx , Lout , Cout , and Ct can be derived by applying several constraints on the systems response to the variable load resistance RL. In this analysis, it is assumed that the components are lossless. In addition, assumptions are made about the class E to allow use of Raab’s equations [17], namely that the transistor is
Figure 2-1. One-to-one wireless power system block diagram.
17 Figure 2-2. Class E driving circuit for a wireless power system.
a perfect switch and that the choke inductance is infinite. Before these derivations it is
necessary to have expressions for receiver impendance, Zrx , input impedance looking
into the transmitter coil, Zin, and impedance looking into Lout , Ztx . Zrx can be expressed as follows:
Zrx = Rrx + jXrx
= RL||Crx (2–1) R − jωC R2 = L rx L 2 2 2 (2–2) 1 + ω RL Crx
Zin can be found by examining the coupling equations:
V1 = jωL1I1 + jωMI2
V2 = jωMI1 + jωL2I2 (2–3)
where L1,L2, and M, are transmitter coil, receiver coil, and mutual inductance, respectively. Zin is V1/I1
18 Zin = Rin + jXin ω2M2R = rx R2 + ( L + X )2 µrx ω 2 rx ¶ ω2M2(ωL + X ) + j L − 2 rx ω 1 2 2 (2–4) Rrx + (ωL2 + Xrx )
Ztx is just Zin with additional series reactance from Cout and Lout :
Z = R + jX tx tx µtx ¶ 1 = Rin + j ωLout − + Xin (2–5) ωCout
2.2.1 Design Equation for Crx
Selection of Crx is determined on the basis of efficiency and quality factor Q. If the
real part of Zin is too low compared to the coil parasitics, the system will be inefficient. If it is too large, it is difficult to get Q high enough for class E operation (about 1.78 [18]).
By forcing the peak real part of Zin to be a specified value (R0), a compromise between efficiency and Q can be reached. To derive which Crx forces the maximum real part of
∂Rin Zin to be R0, the RL corresponding to the peak value is found by setting to zero. This ∂RL yields a polynomial of degree six, where four of the roots are comprised by a double conjugate pair and can thus be ignored
j RL = § (2–6) ωCrx There are two real roots
ωL R = § 2 L 2 (2–7) 1 − ω L2Crx
which can be substituted back into Eq. (2–4); then the real part is set to R0
19 M ωM R = § = R in 2 0 (2–8) 2L2 ω Crx L2 − 1
Solving for Crx yields two roots
R L § 1 ωM2 C = 0 2 2 rx 2 2 (2–9) R0ω L2
The negative root gives a Crx which ensures that Zin phase will increase with
increasing RL, which has the desirable effect of lowering power delivery at high load resistance. This is desirable because in the case of a device being charged, high load resistance (thousands of Ω) corresponds to fully-charged condition and thus low power
requirement.
2.2.2 Design Equation for Lout
The purpose of Lout is to ensure the circuit has a minimum Q high enough for
proper functioning of the class E. Q is smallest when the real part of Zin is highest, at R0.
Since Lout contributes the largest part of the reactance of Ztx ,
ωL Q ∼ out (2–10) R0
Lout is found:
−1 Lout = ω QR0 (2–11)
2.2.3 Design Equation for Cout
Cout brings the range of the phase of Ztx to a range which allows ZVS operation of the class E and maximum efficiency. From [19], this phase range is 40o to 70o . By setting the minimum phase to a specified value, φ, efficient operation can be achieved.
The location of the minimum phase is where
∂∠(Z ) tx = 0 (2–12) ∂RL
which yields a quadratic equation in RL with the roots
20 µ ¶ L 2 R = §2R 2 L 0 M s C ( L (k 2 − 1) − R Q) − 1 × ω out ω 1 0 2 (2–13) ωCout (ωL1(k − 1) − R0(Q + 2)) − 1 q 2 where k = M . Substituting Eq. (2–13) into Eq. (2–5) and setting tan(φ) = Xtx L1L2 Rtx and squaring to eliminate the radical in the numerator yields a quadratic equation in
Cout , with two real roots.
−1 2 Cout = ω {ωL1(1 − k ) + R0(Q + 1 § sec(φ))}
× { 2 2 − 2 − 2 R0 (Q + 2Q tan(φ) ) + 2QR0ωL1(1 k ) 2 2 − 2 2 − 2 }−1 + ω L1(1 k ) + 2ωL1R0(1 k ) (2–14)
The greater root, corresponding to positive φ, yields
ω−1 C = out 2 (2–15) ωL1(1 − k ) + R0(Q + 1 − sec(φ)) after simplification.
2.2.4 Design Equation for Ct
Finally, Ct is selected to guarantee zero voltage switching (ZVS) operation of the class E. From [17], this optimum value of Ct , given a load resistance R is
2ω−1 C = t π2 (2–16) (1 + 4 )R
Since the load resistance in the wireless power system is variable, Ct is selected based on the maximum R, which for the circuit under consideration is the magnitude of
Ztx as RL increases to infinity. Taking this limit,
21 Table 2-1. Design parameters. Parameter Value L1 34.58 µH L2 4.05 µH M 1.65 µH R0 7.5 Ω Q 2 φ 65o
Table 2-2. Component values. Component Calculated Measured Crx 100.55 nF 100.00 nF Cout 11.90 nF 11.57 nF Ct 15.15 nF 14.55 nF Lout 9.95 µH 9.52 µH
1 ωC R = L + − 2M2 rx ω out ω 2 (2–17) ωCout ω Crx L2 − 1 and substituting in the derived values for the other components (Eqs. (2–9), (2–11),
(2–15)), the optimum Ct is found:
2ω−1 C = t π2 (2–18) (1 + 4 )(1 + sec(φ))R0
2.3 Tests
Having derived the optimum component values for a given L1, L2, M, R0, Q, and φ, this section demonstrates the performance of the system.
A test system was built, consisting of a 16 cm by 18 cm, 13 turn, spiral transmitting coil, designed by the technique described in [20], and a rectangular 4 cm by 5 cm,
6 turn, receiving coil. Both coils were constructed of 100 strand, 40 AWG Litz wire to minimize coil parasitics. Fig. 2-3 shows a picture of the test system. Table 2-1 gives values of the inductances L1, L2, M, and design parameters R0, Q, and φ. R0 was chosen as 7.5 Ω based on the total transmitting and receiving coil parasitics which amounted to about 0.5 Ω. In general, selection of R0 is system-dependent but it
22 Figure 2-3. Test setup. Top is a diagram showing the coils, where red is the receiver and blue is the transmitter. Bottom is a photograph of the coils.
23 4 Figure 2-4. From top left, clockwise: Rin, ∠Ztx , drain voltage waveform at RL = 10 Ω, and Q.
should be at least an order of magnitude higher than the parasitics. How large an R0 is acceptable depends on the availability of inductors of sufficient size to achieve minimum Q. Minimum Q and φ values were chosen based on allowable values given in [18, 19], with additional buffer to tolerate some deviations of real components. Based on the proposed method and equations in Section 2.2, the optimum component values were calculated. Table 2-2 gives the calculated component values and also lists the measured values of the actual components being used. Fig. 2-4 shows the calculated Rin, ∠Ztx , drain voltage waveform, and Q, demonstrating that the desired constraints are met using the component selection formulas. One of the key challenges for a wireless power system is to have desirable performance responding to variable load. To evaluate the system performance with regards to power and efficiency, RL was swept from 60 to 4000 Ω by means of an electronic load. The DC voltage and current were measured at the load and at the supply. Supply voltage was 12 V.
24 Figure 2-5. Received power and total efficiency as a function of RL. Circles are measured values and the solid line is simulated, using the calculated component values.
Fig. 2-5 shows the received power and total efficiency versus load resistance, and the simulated values of power and efficiency, using the ideal, calculated component values. The system has power delivery of over 3.7 W, and peak efficiency over 66%. The important feature in Fig. 2-5 is the trend of decreasing received power and total efficiency with increasing RL, which is guaranteed by the component selection. The ideal performance is close to the real system. Power and efficiency are lower by 5%-10% in the actual system, primarily due to the deviation of Lout and Cout in the real components. Deviations from their ideal values cause the φ and Q to shift which has a substantial effect on the class E efficiency.
To further investigate the sensitivity to component selection, a Monte Carlo simulation was run, assuming the components are normally distributed, with means given by the derived component formulas and with standard deviations, σ, such that 3σ is the component tolerance. These simulations were carried out at tolerance levels of
25 Figure 2-6. Received power and total efficiency as a function of RL, and their 95% confidence intervals with 5% component tolerance (circles), 10% component tolerances (pluses), and 20% component tolerances (squares).
5%, 10%, and 20%. Fig. 2-6 shows the 95% confidence intervals for received power and total efficiency at the three tolerance levels. As can be seen, the power is skewed low, with tolerances of about +35/-25% at 5% tolerance, +115/-45% at 10% tolerance, and +425/-70% at 20% tolerance. The efficiency is skewed very high, with tolerances of about +3.4/-18% at 5% tolerance, +3.6/-38% at 10% tolerance, and +4.3/-70% at 20% tolerance. This skew low in the power confidence intervals and skew high in the efficiency confidence intervals shows that the system is not optimized for maximum power delivery but rather efficiency. This makes sense, as all of the constraints (R0, φ, Q, and ZVS) used for component selection are chosen to maximize the efficiency of the
class E. At tolerance of 5% or less, the confidence interval shows that the performance is still decent. For greater tolerances, the potential variability of performance is probably unacceptable for most applications.
26 Figure 2-7. Efficiency as a function of RL. Squares are class E efficiency, pluses are coupling efficiency, and circles are total efficiency.
Fig. 2-7 shows the amplifier efficiency, coupling efficiency, and total efficiency.
Amplifier efficiency is AC transmitting power over DC input power; coupling efficiency is DC power at the load over AC transmitted power; and total efficiency is the product of these two efficiencies. From Fig. 2-7, the coupling efficiency comprises most of the losses. The coupling efficiency decreases with increasing load resistance because as
RL increases, Rin decreases, while the parasitics remain the same, so more power is dissipated through the parasitics. The amplifier efficiency peaks where ∠Ztx is in the 40o − 70o range, which by design is in the neighborhood of the phase minimum. 2.4 Conclusion
This chapter has presented a set of design equations for optimizing the performance
of a class E amplifier used in inductively coupled wireless power system. By applying constraints on the real part of the input impedance to the primary coil, the phase of the input impedance, the minimum Q, and the drain voltage waveform, components can
be selected to guarantee desirable operating characteristics of the system, namely, the
27 ZVS operation of the class E and the trend of decreasing power received with increasing load resistance. The proposed optimization method was tested in a system composed of a 16 cm by 18 cm primary coil and a 4 cm by 5 cm secondary coil with a variable load. The system shows power delivery of over 3.7 W, and peak efficiency over 66%, in addition to the desirable trend of decreasing power and efficiency with respect to increasing load resistance.
28 CHAPTER 3 NEAR-FIELD ELECTROMAGNETIC ANALYSIS
3.1 Introduction
The performance of a near-field wireless power system depends heavily on the electromagnetic properties of the primary and secondary coils. In particular, the fields produced by the coil, the coil inductance, and the coil parasitics are important to know. This chapter derives analytically these quantities for round and rectangular conductors for use in system design.
3.2 Coil Fields
At frequencies less than 500 kHz, instead of solving Maxwell’s coupled equations, it is still sufficiently accurate to calculate the magnetic field using the magnetostatic solution, that is, the Biot-Savart Law. This is known as the magnetoquasistatic (MQS) solution [21]. The analytical MQS solution for a line of current is presented here (see
Fig. 3-1). The fields produced by a polygonal coil can be constructed by superposition.
Z 0 1 ~J(~r ) ׈i 0 H = r r dV 0 0 2 (3–1) 4π 0 |~r − ~r | ZV I ξc ~c × ~adξ = (3–2) 4π |a|(ξ2 + r 2)3/2 ξb 0 ¯ I ~c × ~a ξ ¯ξc = ¯ (3–3) | | 2 2 2 1/2 4π a r0 (ξ + r0 ) ξb ³ ´ I ~c × ~a ~a · ~c ~a · ~b = − (3–4) 4π |~c × ~a|2 |c| |b|
To test the MQS analytical solution, it was compared to a method of moments (MoM) solution [22] calculated using the Numerical Electromagnetics Code (NEC) [23].
In particular, a single turn, 1 m by 1 m square coil was tested using both techniques. The fields were calculated at a 2m by 2m plane 5 cm above the coil. While frequency is not used in MQS, it is in MoM, and the frequency used in this case was 240 kHz.
29 Figure 3-1. Current stick for MQS analysis.
Fig. 3-2 shows the magnitude of the the field components (Hx ,Hy ,HZ ) for both MQS and MoM techniques. From the plots, it is evident that the distributions are similar. The magnitude of the difference, in terms of root mean square difference of the x-component is 6.3239×10−5 A/m, of the y-component is 9.4997×10−5 A/m, and of the z-component is 8.4331×10−5. The peak field magnitude in both cases is 1 A/m. 3.3 Coil Inductance
The inductance matrix Mij relates the flux from a primary coil i through a secondary coil j to the primary’s current. [24] presents many inductance formulas for different geometries. The most general for a filamentary coil is the Neumann formula:
I I µ ds~ · ds~ M = 0 i j (3–5) ij 4 ~ π ci cj |Rij | The self inductance is similar: I I ¯ µ0 ds~ i · ds~ i ¯ Mii = ¯ + Lp,ii (3–6) 4 ~ |R|≥a/2 π ci ci |Rii |
where Lp,ii is the internal inductance of the conductor and a is the radius.
30 Figure 3-2. Magnetic field components using MQS and MoM techniques of a 1m by 1m square coil.
Figure 3-3. Magnetic field magnitude using MQS and MoM techniques of a 1m by 1m square coil.
31 3.4 Coil Parasitics
The effect of the distribution of current with in the conductor is magnified at higher
frequency. Internal resistance and inductance depend on the distribution of current. There are two main mechanisms for high frequency losses and internal inductance: the skin effect and proximity effect. The skin effect is when current distribution is mostly towards the surface of the conductor, raising the resistance. Proximity effect is when magnetic fields from nearby conductors induce eddy currents on the conductor surface, increasing the power dissipation. These two effects are discussed in the context of transformer design in the literature; there are analytical [25, 26] and empirical
[27, 28] treatments of these effects. However, the results published in the literature
are incomplete in that the exact current distributions are not given, and neither is the effect on conductor internal inductance. In addition, no paper derives these effects for a conductor of arbitrary rectangular cross section, such as a PCB trace. This section derives the current distribution and resistance and inductance under skin and proximity effects for both round and rectangular conductors. 3.4.1 Round Conductor
[29] presents a partial description of the skin effect in round conductor. This section
presents a complete description of skin and proximity effects. Both skin and proximity effect parasitics for a round conductor are derived through application of Maxwell’s equations. 3.4.1.1 Skin effect
Fig. 3-4 shows the wire cross section and boundary conditions. Using the current distribution equation, separation of variables, and the surface electric field and symmetry
32 Figure 3-4. Conductor cross section showing field and current in round conductor under skin effect.
as boundary conditions, the total current can be related to the surface electric field E0:
2 ∇ Jz = jωσµJz (3–7)
α2 = −jωσµ (3–8) ∂2J 1 ∂J z + z + α2J = 0 (3–9) ∂r 2 r ∂r z
Jz = C1J0(αr) (3–10)
J0(αr) Jz = σE0 (3–11) J0(αr0) J0(αr) Ez = E0 (3–12) J0(αr0) 1 ∂E H = z (3–13) φ jωµ ∂r σE J0 (αr) = 0 0 (3–14) I α J0(αr0) ~ ~ H · dl = I = 2πr0Hφ(r0) (3–15) σE J0 (αr) I = 0 0 (3–16) α J0(αr0) where Jz is current, Hφ is φ-directed magnetic field, Ez is z-directed electric field, and I is total current.
33 Figure 3-5. Conductor cross section showing field and current in round conductor under proximity effect.
The next steps relate the total current to the parasitics. With the definition of internal impedance,
E (r ) Z = z 0 (3–17) I = R + jωL (3–18) 1 R = (3–19) s σδ 1 δ = √ (3–20) πf σµ √ r γ = 2 0 (3–21) δ ³ 0 0 ´ Rs Ber(γ)Bei (γ) − Bei(γ)Ber (γ) Rp = √ (3–22) 2πr Ber 02(γ) + Bei 02(γ) 0 ³ ´ R Ber(γ)Ber 0(γ) − Bei(γ)Bei 0(γ) L = √ s p 02 02 (3–23) 2πr0 Ber (γ) + Bei (γ) where Bei and Ber are the imaginary and real Kelvin functions.
3.4.1.2 Proximity effect
Similarly, proximity effect can be handled by applying the boundary condition of an imposed tangential surface magnetic field. Fig. 3-5 shows the wire cross section and boundary conditions. Beginning with the current distribution equation, the fields can be related to the real and reactive power. The power terms may be related to the parasitics
34 through the total current: µ ¶ 1 ∂2J 1 ∂ ∂J z + r z + α2J = 0 (3–24) r 2 ∂θ2 r ∂r ∂r z
Jz = C1J1(αr) cos(θ) (3–25)
J1(αr) Jz = H0 cos(θ) (3–26) J1(αr0) H0 J1(αr) Ez = cos(θ) (3–27) σ J1(αr0)
jωσµH~ = ∇ × Ez (3–28)
³ ´ H ˆr J (αr) J0 (αr) H~ = 0 1 sin( ) + ˆ 1 cos( ) 2 θ θα θ (3–29) α r J1(αr0) J1(αr0)
Z Z σ 2π r0 P = |E|2rdrdθ (3–30) real 2 0Z 0Z ωµ 2π r0 P = |H|2rdrdθ (3–31) imag 2 0Z ¯0 ¯ 2 r0 2 π H0 ¯ J1(αr) ¯ Preal = ¯ ¯ rdr (3–32) 2 σ 0 J1Ã(αr0) ! Z ¯ ¯ ¯ ¯ ωµπ H2 r0 1 ¯ J (αr) ¯2 ¯ J0 (αr) ¯2 P = 0 ¯ 1 ¯ + ¯ 1 ¯ rdr imag 2 2 (3–33) 2 α 0 (αr) J1(αr0) J1(αr0) 2P R = real p 2 (3–34) Ip 2P L = imag p 2 (3–35) ωIp
3.4.2 Rectangular Conductor
A rectangular conductor of width w and thickness t, such as a printed circuit board
(PCB) trace can be handled with the current distribution PDE in Cartesian coordinates. Similar derivations as in the previous section can be carried out in order to derive the parasitics for a rectangular cross section conductor. The following sections derive the parasitics through application of Maxwell’s equations.
35 Figure 3-6. Conductor cross section showing field and current in rectangular conductor under skin effect.
3.4.2.1 Skin effect
Fig. 3-6 shows the rectangular cross section and boundary conditions used in this section. The boundary condition is one of a constant tangential electric field. Using the reactive and real powers within the cross section, and the total current, the parasitics may be derived. Applying the PDE and separation of variables, where the total current distribution must be handled by superposition of two solutions, one associated with a homogeneous x-direction (Jz1) and another associated with a homogeneous y-direction
(Jz2):
36 2 2 ∇ Jz = −α Jz (3–36) ∂2J ∂2J z + z + α2J = 0 (3–37) ∂x 2 ∂y 2 z
Jz = Jz1 + Jz2 (3–38)
Jz1 = X (x)Y (y) (3–39)
X (x) = A cos(βx) + B sin(βx) (3–40)
X 0(0) = 0 (3–41)
X (w/2) = 0 (3–42)
X = A cos(βnx) (3–43) π β = (2n + 1) (3–44) n w
Y (y) = C cosh(γy) + D sinh(γy) (3–45)
2 2 − 2 γn = βn α (3–46)
Y 0(0) = 0 (3–47)
Y (t/2)X (x) = σE0 (3–48)
Y = a cosh(γ y) (3–49) n n R n w/2 0 σE0 cos(βnx)dx an = R (3–50) w/2 2 cosh(γnt/2) 0 cos (βnx) 4(−1)n an = (3–51) π(2n + 1) cosh(γnt/2) ∞ Jz1 = σE0Σ0 an cosh(γny) cos(βnx) (3–52)
37 Jz2 = X (x)Y (y) (3–53)
Y (y) = A cos(δy) + B sin(δy) (3–54)
Y 0(0) = 0 (3–55)
Y (t/2) = 0 (3–56)
Y = A cos(δny) (3–57) π δ = (2n + 1) (3–58) n t
X (x) = C cosh(²x) + D sinh(²x) (3–59)
2 2 − 2 ²n = δn α (3–60)
X 0(0) = 0 (3–61)
X (w/2)Y (x) = σE0 (3–62)
X = b cosh(² x) (3–63) n n R n t/2 0 σE0 cos(δny)dy bn = R (3–64) t/2 2 cosh(²nw/2) 0 cos (δny) 4(−1)n bn = (3–65) π(2n + 1) cosh(²nw/2) ∞ Jz2 = σE0Σ0 bn cosh(²nx) cos(δny) (3–66)
38 4(−1)n J = σE Σ∞ (3–67) z 0 0 π(2n + 1) ³ ´ cosh(γ y) cos(β x) cosh(² x) cos(δ y) × n n + n n (3–68) cosh(γnt/2) cosh(²nw/2) ∂J −jωµσH = z (3–69) x ∂y ∂J jωµσH = z (3–70) y ∂x σE 4(−1)n H = 0 Σ∞ (3–71) x α2 0 π(2n + 1) ³ ´ γ sinh(γ y) cos(β x) δ cosh(² x) sin(δ y) × n n n − n n n (3–72) cosh(γnt/2) cosh(²nw/2) σE 4(−1)n H = 0 Σ∞ (3–73) y α2 0 π(2n + 1) ³ ´ β cosh(γ y) sin(β x) ² sinh(² x) cos(δ y) × n n n − n n n (3–74) cosh(γ t/2) cosh(² w/2) Z Z n n I t/2 t/2 = J dxdy (3–75) 4 z 0 0 ³ ´ 16(−1)n I = σE Σ∞ tanh(γ t/2)w + tanh(² w/2)t (3–76) p 0 0 π(2n + 1) n n Z Z σ 2π r0 P = |E|2rdrdθ (3–77) real 2 0Z 0Z 2π r0 ωµ 2 Pimag = |H| rdrdθ (3–78) 2 0 0 2P R = real p 2 (3–79) Ip 2P L = imag p 2 (3–80) ωIp
3.4.2.2 Proximity effect
Fig. 3-7 shows the rectangular cross section and boundary conditions used in this section. The boundary condition is one of an arbitrary, constant, tangential, magnetic field. Using the reactive and real powers within the cross section, and the total current, the parasitics may be derived. Proximity effect can be handled with the imposition of
39 Figure 3-7. Conductor cross section showing field and current in rectangular conductor under proximity effect. tangential magnetic field on the vertical sides, and odd symmetry about the y-axis:
2 2 ∇ Jz = −α Jz (3–81) ∂2J ∂2J z + z + α2J = 0 (3–82) ∂x 2 ∂y 2 z
Jz = X (x)Y (y) (3–83)
Y (y) = A cos(γy) + B sin(γy) (3–84)
Y 0(0) = 0 (3–85)
Y (t/2) = 0 (3–86)
Y = A cos(γny) (3–87) π γ = (2n + 1) (3–88) n t
X (x) = C cosh(βx) + D sinh(βx) (3–89)
2 2 − 2 βn = γn α (3–90)
X (w/2)Y (x) = σE0 (3–91)
Xn = an sinh(βnx) (3–92)
40 ¯ 2 ∂Jz ¯ α Hoy = − ¯ (3–93) ∂x x=w/2 R − 2 t/2 α H0y 0 σE0 cos(γny)dy an = R (3–94) t/2 2 βn cosh(βnw/2) 0 cos (γny) 2 n+1 α H0y 4(−1) an = (3–95) π(2n + 1)βn cosh(βnw/2) ∞ Jz = Σ0 an sinh(βnx) cos(γny) (3–96)
Z Z I t/2 t/2 = J dxdy (3–97) 4 z 0 0 ³ ´ 16(−1)n I = σE Σ∞ tanh(γ t/2)w + tanh(β w/2)t (3–98) p 0 0 π(2n + 1) n n Z Z σ 2π r0 P = |E|2rdrdθ (3–99) real 2 0Z 0Z 2π r0 jωµ 2 Pimag = |H| rdrdθ (3–100) 2 0 0 2P R = real p 2 (3–101) Ip 2P L = imag p 2 (3–102) ωIp
3.5 Litz Wire
To mitigate the high frequency parasitics described in the previous sections, Litz
wire could be used as the coil conductor [30]. Litz wire is stranded wire where the strands are insulated from one another. Since the size of the individual conductors is much less than the skin depth, the skin and proximity effects are minimized. Typically, Litz wire is specified in terms of the number of strands and the gauge of the individual wires. For instance, “100/40” Litz wire is 100 strands of 40 AWG wire. 3.6 Regulations
As the application of wireless power considered in this dissertation is largely
consumer electronics, some discussion of the health and human safety aspects, as well as appropriate federal regulations, is in order. This section summarizes some regulatory constraints.
41 FCC Part 18 [31] concerns unlicensed intential, unintential, or incidental radiators for industrial, medical, or scitific (ISM) and non-ISM equipment. According to Part 18, operation within specific search and rescue bands (490-510 kHz, 2170-2194 kHz, 8354-8374 kHz, 121.4-121.6 MHz, 156.7-156.9 MHz, and 242.8-243.2 MHz). Additionally there are field strength limits for different frequency bands and applications and conducted emission limits. If a wirelss device is designed to work in non ISM bands the the field strength limit is 15 µV/m at 300 m and the conducted emission limit is 66-56 dBµV (quasi-peak) and 56-46 dBµV (average)
FCC Part 15 [32] is concerned with radio frequency devices. Specifically, subpart B concerns unintentional radiators. There are radiated and conducted emission limits. For conducted emissions, the limit is 66-56 dBµV (quasi-peak) and 56-46 dBµV (average).
Radiated emission limits are (in µV/m), for frequency between 9 and 490 kHZ, 2400 divided by the frequency in kHz, measured at 300 m. For frequency between 490 and 1705 kHZ, 24000 divided by the frequency in kHz, measured at 30 m. IEEE C95.1 describes limits on field strength, current and specific absorption rate (SAR) for health and human safety [33]. Exposure limits are defined for two cases, controlled and uncontrolled environments (the general public). For consumer electronics, and near-field induction, the primary concern would be magnetic field strength limits in uncontrolled environments. Maximum permissible exposures (MPEs), for head and torso, in uncontrolled environments, between 3.35-5000 kHz, are an rms flux of 0.205 mT and field of 163 A/m. In the limbs, MPEs are an rms flux of 1.13 mT and field of 900 A/m. Additionally, there are specific absorption rate (SAR) limits: for the whole body, 0.08 W/kg; for any localized 10 g of tissue, 2 W/kg; and for extremities, 4 W/kg. 3.7 Conclusion
This chapter analytically derived useful relationships for the fields produced by the coil, the coil inductance, and the coil parasitics for round and rectangular conductors.
42 These relationships can be used to help estimate performance and guide the design of the coils before construction of a system. Combined with the design equations from Chapter 2, this provides a basis for an electronic design automation (EDA) code used throughout the rest of this dissertation.
43 CHAPTER 4 OPTIMAL PRIMARY COIL DESIGN
4.1 Introduction
If multiple devices are to be charged simultaneously on the same system, the transmitting coil must be large enough to accomodate them. This poses a challenge, as to ensure uniform power delivery to devices, regardless of position, the electromagnetic field distribution must be even. In particular, the distribution of the z-component of the magnetic field in the plane of the receiving coils must be as uniform as possible. Transmitting coils may be designed to produce such fields; one approach is the optimal hybrid coil design [34], which is demonstrated for as large a coil as 15 cm by 15 cm.
This chapter describes a different technique for coil design (the primary difference being the parameterization of the coil shape), which is demonstrated for a 20 cm by 20 cm coil.
4.2 Planar Wireless Power System
In this chapter, the system was configured with series-parallel compensation as described in [13]. The transmitting coil follows, which is in turn inductively coupled to the receiving coil, a rectangular coil of 6 cm by 8 cm and 6 turns. Both transmitter and receiver coils were constructed by hand using Litz wire to reduce resistive losses from proximity and skin effects. The receiving coil is connected to the second half of the
Figure 4-1. Transmitter test setup.
44 20
18
16
14
12
10
8 y position (cm)
6
4
2
0 0 2 4 6 8 10 12 14 16 18 20 x position (cm)
Figure 4-2. Coil layout. transformation network, a parallel capacitor, and followed by a rectifier and a receiver load. A picture of the test setup is shown in Fig. 4-1.
4.3 Coil Design
The transmitting coil is a rectangular spiral with blunted corners, where the ratio of the width of a turn to the overall width, f , is defined by:
f = 1 − (1 − (N − n + 1)/N)k (4–1)
where n is the turn number, counting from the outside, and N is the number of turns, k is a parameter, and ∆ gives the fraction of of each corner to be removed to blunt the corners. The spiral geometry is entirely described by free parameters N, k, and ∆; and the length and width, which are fixed at 20 cm by 20 cm for this example. By sweeping the parameter values, evaluating the fields, and calculating an objective function, the
45 20
250 18
200 16
150 14
100 12
10 50
8 0 y position (cm)
6 −50
4 −100
2 −150
0 0 2 4 6 8 10 12 14 16 18 20 x position (cm)
Figure 4-3. Calculated z-directed magnetic field, assuming 1 A current (A/m). coil design which produced maximally flat fields for a coil of the specified size was determined. The analytical magnetoquasistatic (MQS) solution for a line of current (see Chapter 3) was used to build the fields for the entire current in a plane 1 mm above the coil. The objective function was chosen as the coefficient of variation (COV, the standard deviation divided by the mean) of the z-component of the magnetic field. Minimizing the COV minimizes the relative variations in the field, ensuring a smooth distribution. The final optimal coil layout is shown in Fig. 4-2, and the corresponding MQS fields are shown in Fig. 4-3.
4.4 Testing
The transmitting coil was tested in three ways. First, the z-directed magnetic field was measured with a 6 cm diameter field probe. Second, the receiver position was varied over the entire transmitter coil area to gauge the uniformity of wireless power transfer. Finally, the receiver was fixed at the center of the transmitter and the load resistance was swept from 10 to 2000 Ω.
46 20
18 200
16
180 14
12 160
10
8 140 y position (cm)
6
120 4
2 100
0 0 2 4 6 8 10 12 14 16 18 20 x position (cm)
Figure 4-4. Field probe measurement (mV).
The chief figures of merit considered in the test were the DC power supplied to the amplifier, the AC power transmitted, and the DC power delivered to the resistive load, in addition to the amplifier, the coupling, and the total efficiency. Amplifier efficiency is defined as the ratio of transmitted power to supplied power; coupling efficiency is defined as the ratio of power received by the load to transmitted power; and total efficiency is defined as the product of the previous two.
4.5 Results
Table 4-1 summarizes the perfomance charcteristics of the coil. The field measurement results are shown in Fig. 4-4, in terms of the voltage measured on the field probe. The peak in the lower left corner corresponds to the location of the input leads, and the peaks in other corners are due to the superposition of fields at corners of the spiral. The apparent drop-off at the edges is due to the spatial averaging effect of the probe. A small portion of the probe was outside of the coil, where the field reverses and
47 Table 4-1. Summary of system performance. Size 20 cm by 20 cm Peak delivered power 11.8 W Peak total efficiency 80.9% Peak coupling efficiency 88.4% COV 2.2% Self-inductance 45.00 µH Resistance 0.37 Ω
becomes negative, pulling the average down. Aside from this artifact of the field probe
measurement, the general trend of the field matches the MQS calculations. From Fig. 4-5, the spatial uniformity of the received power is shown. The two
notable peaks match the field peaks at the corners and near the leads, which can be seen in the field plots. These peaks are relatively small, however, as the maximum variation shown in the plot is 0.8 W, less than 10% of the mean. The COV is likewise small, at 2.2%.
Fig. 4-6 shows the results from the variable loading test. As can be seen, the received power is maximum at 25 Ω and a value of 11.8 W. The maximum total efficiency is 80.9% at 100 Ω and the maximum coupling efficiency is 88.4% at 250
Ω. The efficiencies are high under a wide range of loads (it should be noted that the efficiency is lessened slightly in the presence of multiple loads). This demonstrates the system’s robustness, not only with respect to receiver placement, but also loading conditions. 4.6 Conclusion
This chapter has demonstrated the feasibility of large transmitting coils for open air inductively coupled power transfer. Large transmitting coils such as this may be used for wireless charging of multiple battery-powered devices equipped with receiving coils, such as cellphones and PDAs. The primary challenge in designing such a coil is achieving an even power delivery regardless of receiver position, in order to accommodate multiple devices. Such a coil design was achieved through optimization, and the 20 cm by 20 cm coil was built and tested with a switchmode power amplifier
48 20
9.3 18
16 9.2
14 9.1
12 9 10
8.9 8 y position (cm)
6 8.8
4 8.7
2 8.6 0 0 2 4 6 8 10 12 14 16 18 20 x position (cm)
Figure 4-5. Received power (W) as a function of the location of the center of the receiving coil.
20 Input power Transmitter power 15 Recieved Power
10 Power (W) 5
0 1 2 3 4 10 10 10 10
100
80
60 Coupling efficiency
Efficiency (%) 40 Amplifier efficiency Total efficiency 20 1 2 3 4 10 10 10 10 Rl (Ω)
Figure 4-6. Power (W) and efficiency (%) at loads from 10 Ω to 2 kΩ.
49 and a 6 cm by 8 cm receiving coil. It was found to have a maximum efficiency of 80.9% and a maximum power delivery of 11.8 W. At a fixed load, the power delivery has a coefficient of variation of 2.2% as the receiving coil’s position is varied on the transmitter, and the peak spatial variation is less than 10% of the mean power delivery. In general, the system is robust and efficiency is high, irrespective of receiver placement and loading conditions. This demonstrated the feasibility of eliminating the last wire of wireless portable devices to achieve a completely wireless solution.
50 CHAPTER 5 M:N ANALYSIS
5.1 Introduction
Regardless of technique, larger transmitting coils require more turns to achieve an
even field distribution, raising the inductance. This is a problem because the amplifier operation is sensitive to component variation in the transformation network following the driving circuit. As the inductance of the primary coil increases, the series capacitor in the network needs to be smaller, and the class E becomes increasingly sensitive to small variations in the component values, sometimes severely hindering system performance. To circumvent this problem, the inductance could be lowered by using two or more primary coils in parallel. This reduces the inductance while still allowing a large charging area. In addition, having multiple tansmitting coils in parallel reduces the influence of one load’s power consumption on that of any other load. This chapter derives and verifies the mathematical description of the coupling between M transmitters and N receivers and demonstrates the advantages of such a system experimentally.
5.2 Analysis
The mathematical analysis of power transfer in the M:N case can be performed by
applying Kirchoff voltage and current laws to the circuit shown in Fig. 5-1. The primary coils are numbered 1 through M, and the receiving coils are numbered M + 1 through
M + N. The voltage-current matrix equation is:
ZI = V (5–1)
b=XM+N ZabIb = Va (5–2) b=1 th th where Ib is the current on the b coil and Va is the voltage on the a coil, and Zab is the (a, b)th element of the impedance matrix, defined as
51 Figure 5-1. M:N block diagram.
jωLa + Ra for a = b Zab = (5–3) jωMab otherwise
where ω is angular frequency, La and Ra are the self-inductance and parasitic
th th th resistance of the a coil, and Mab is the mutual inductance between the a and b coils. Relating current and voltage in each of the coils, Vb can be found. For the primary coils (in parallel), the voltage is the same for all, the input voltage (Vb = Vin). For coils
M + 1 through M + N, Vb = IbZLb, where ZLb is the impedance of the load and any transformation network attached to the bth coil. The final constraint is that the sum of the currents in the primary coils must be equal to the input current, Iin = ZinVin. Applying this to Eq. 5–2,
(Z − ZL)I = 0 (5–4)
where Z is defined as before, I is a vector of the currents, and ZL is defined as
52 Z 1 ≤ a ≤ M 1 ≤ b ≤ M in for and Z = − L ZLb for a = b and b > M (5–5) 0 otherwise
Eq. 5–4 can be solved for Zin by splitting it into several submatrices as follows: ZIII (ZII )T Z − ZL = (5–6) ZII ZI III I = (5–7) II
where ZIII has dimensions M × M; ZII has dimensions N × M; ZI has dimensions
II I IV III N×N; I has dimensions M×1; and I has dimensions N×1. Defining Z = Z +Zin1MM
(where 1MM is an M × M matrix of ones), and with some manipulations,
ZI II = −ZII III (5–8)
IV II II T I (Z − Zin1MM )I = −(Z ) I (5–9)
Input current Iin is the sum of currents in the transmitting coils, stated mathematically
as (where 11M is a 1 by M vector of ones):
I Iin = 11M I (5–10)
Using Eq. 5–8 through 5–10,
IV II T I −1 II II II [Z − (Z ) (Z ) Z ]I = Zin1MM I (5–11)
II Substituting Vin = Zin1MM I and using Zin = Vin/Iin:
53 IV II T I −1 II −1 Zin = 1/{11M [Z − (Z ) (Z ) Z ] 1M1} (5–12)
Having a closed-form expression for the input impedance allows derivation of the
II currents in the individual coils. By subtracting Zin1MM I from both sides of Eq. 5–11:
III = null(X) (5–13)
IV II T I −1 II X = Z − (Z ) (Z ) Z − Zin1MM (5–14)
II = −(ZI )−1(ZII III ) (5–15)
Now knowing the currents in the transmitter and receiver coils, the power received II 2 by load b may be computed simply as ( b) Re(ZLb). These equations are extensible to different receiver topologies, such as parallel or series capacitors, and nonlinearities (such as rectifiers, or proximity and skin effects on resistance and inductance) may be considered as well, through the use of fixed-point iteration.
5.3 Tests Results
To verify the correctness of the preceding equations as well as to demonstrate the benefit of using multiple primary coils in parallel, simulations and tests were carried out for the 1:1, 1:2, 1:3, 2:2, and 2:3 cases. For all except the three-receiver cases, two receiver sizes were considered. In addition, the two-transmitter tests were performed with the transmitting coils adjacent and separated. Fig. 5-2 shows the eleven different configurations for the test setup.
The primary coil inductance is 34.44 µH, reduced by half when the two-coil case is considered. Component selection procedure for the class E was described in [16], and component values are specified in Table 5-1 (for all cases Ldc was 500 µH and
Lout was 9.5 µH). Notably the values for Cout are higher with the 2 transmitter system.
54 Figure 5-2. Starting top row, left-to-right: coil arrangements (thick red line is receiver, thin blue line is transmitter), for (a) 1:1 small-rx, (b) 1:2 small-rx, (c) 1:3 small-rx, (d) 2:2 small-rx, (e) 2:3 small-rx, (f) 1:1 big-rx, (g) 1:2 big-rx, (h) 2:2 big-rx, (i) 2:2 split-tx small-rx, (j) 2:3 split-tx small-rx, and (k) 2:2 split-tx big-rx.
55 Table 5-1. Component values for 1 and 2 transmitter systems.
M Rx size Crx (nF) Cout (nF) Ct (nF) 1 Small 100 11.5 14.7 1 Large 53.8 15.3 6.8 2 Small 100 22.3 27.3 2 Large 53.8 22.3 18.3
Higher capacitance means the impedance will be less sensitive to component variations because of the inverse relationship between capacitance and reactance. The derivative of reactance with respect to capacitance goes as the inverse square of capacitance, so higher capacitance values means a much lower sensitivity. To mitigate proximity and skin effects, we used Litz wire for coil windings. The small receivers were all 4 cm by 5 cm rectangular coils of 6 turns, the large receivers were 7 cm by 8 cm with 6 turns, and the transmitters were 16 cm by 18 cm with 13 turns, designed by the technique described in [20].
For each transmitter/receiver pairing, the resistive load attached to each receiver was swept from 60 Ω to 4000 Ω by means of programmable electronic loads. The resistive load is an approximation of the charge status of a battery; a fully charged device appears as a large resistive load (thousands of Ω) and an uncharged device appears as a low resistive load (a handful of Ω). DC received power (Prx ) flow was measured at the electronic loads. 5.3.1 Verification
To verify the accuracy of the equations developed in Section 5.2, simulations were performed using MATLAB code implementing the analytical treatment of the class E amplifier by Raab [17] for a load with impedance defined as in Eq. 5–12. La and Mab are caluculated using a numerical integration of the Neumann formula [24]. The measured and predicted Prx for each of the M:N cases considered in this chapter are shown in Fig. 5-3. The predicted vs. observed plots show a one-to-one correspondence, aside from some spread due to uncertainty in secondary and primary coil positions. For 1:3 there is a partiularly large amount of spread. With three receivers in close proximity to
56 Figure 5-3. Measured vs. predicted Prx , for: (a) 1:1 small-rx, (b) 1:2 small-rx, (c) 1:3 small-rx, (d) 2:2 small-rx, (e) 2:3 small-rx, (f) 1:1 big-rx, (g) 1:2 big-rx, (h) 2:2 big-rx, (i) 2:2 split-tx small-rx, (j) 2:3 split-tx small-rx, and (k) 2:2 split-tx big-rx. Scale is as indicated in (i) for all subplots. each other, uncertainties in their relative positions have a more pronounced effect on predicted power. 5.3.2 Receiver Decoupling
To show that having multiple primary coils reduces the influence of one receiver on the others, we map the loading condition (Rl1, Rl2, ..., RlN ) to a corresponding received power delivery condition (Prx1, Prx2, ..., PrxN ), using the data from the electronic load sweeps. Though it is impossible to fully explore the power delivery space due to the discrete nature of the tests, looking at this discrete set of loading conditions allows us to outline the physically realizable power values that can be received by multiple loads on the same primary coil or coils.
Figs. 5-4 and 5-5 shows this for the two receiver condition. In Fig. 5-4, 1:2 and
2:2 show similar power spaces because the receivers are small and further apart so they are weakly coupled. Fig. 5-5 demonstrates that when the receiver size is large, for
57 Figure 5-4. Power space plots for two-receiver tests with small receivers.
Figure 5-5. Power space plots for two-receiver tests with large receivers.
58 1:2, the power space is squeezed into a much narrower area, while for 2:2 the power space is close to a square 10 W on each side. The constricted power space for 1:2 occurs because when one load is large (eg, a fully charged device), it “chokes” power delivery to the other, low load (eg, an uncharged device). This phenomenon can be seen in the blue dots (1:2) in Fig. 5-5: when receiver 1 has high load resistance and receives low power (less than 0.2 W), receiver 2 is limited to less than 0.2 W. This amounts to the pinched shape of the power space. Such power delivery limitations are unacceptable. The same plot demonstrates that for 2:2, the power delivered to receiver 2 can still reach about 10 W when receiver 1 has low power, high resistance conditions. Though a simplification, it can be said that with multiple transmitters, the receivers are essentially in parallel while with one transmitter they are essentially in series. With a constant voltage source, power delivery to resistive loads in series is governed by the total resistance, while loads in parallel receive independent power delivery. Multiple primary coils parallelizes power delivery. In the same plots, the effect of split transmitter is also demonstrated. The key
difference for the split transmitter is a reduction in received power, seen as a shifting of the power space towards the origin. This is because the fringing fields of the primary coils dissipate into the nearby environment instead of into a neighboring coil.
Fig. 5-6 shows the power space with small receivers for 1:3 and for 2:3 (large receivers could not be considered for 1:3 because of insufficient room on the transmitter). Though the difference is less pronounced than that of the N = 2 condition, it is apparent that the 1:3 power space is more curved, with an upward sweep, while the 2:3 power space is a distinct rectangular prism. When one receiver is in a high resistance, low power condition, the power received by the other receivers is less in 1:3 than in 2:3. Fig. 5-6 similarly demonstrates the decoupling effect, only with a split transmitter. The effect is the same as discussed in the preceding paragraph, and for similar reasons.
59 Figure 5-6. Power space plot for three-receiver test.
Figure 5-7. Power vs. efficiency plot for two-receiver tests with small receivers.
60 Figure 5-8. Power vs. efficiency plot for two-receiver tests with large receivers.
Figure 5-9. Power vs. efficiency plot for three-receiver tests with small receivers.
61 Table 5-2. Maximum Prx and maximum ηc for different M:N arrangements.
Arrangement Max Prx (W) Max ηc (fraction) 1:2, small-rx 3.44 0.75 2:2, small-rx 3.88 0.75 2:2, small-rx, split-tx 2.60 0.68 1:2, large-rx 1.82 0.82 2:2, large-rx 9.45 0.88 2:2, large-rx, split-tx 7.86 0.87 1:3, small-rx 1.91 0.74 2:3, small-rx 3.08 0.74 2:3, small-rx, split-tx 2.40 0.67
5.3.3 Impact on Efficiency and Total Received Power
Transmitted power was measured using a current probe (Agilent N2783A), a voltage
probe (Agilent N2863A), and an oscilloscope (Agilent DSO 5034A), with a measurement accuracy of 1% and 0.5%, respectively. This corresponds to an accuracy of power measurement of 1.5%. Due to temperature effects and the effect of transmission delay on the phase of measurement, the actual accuracy is estimated to be around 5%. Received power was measured using the DC electronic loads (BK 8500), which have a (worst-case) accuracy of 0.4% for current and 0.38% for voltage, giving a measurement accuracy for power of about 0.8%.
Fig. 5-7 shows total received power, Prx , and coupling efficiency (ηc , defined as the total received power over the transmitted power) for the 2 small receiver tests. It’s clear from the plot that the impact on efficiency is minimal; the maximum ηc for 1:2 and 2:2 is 0.75 and drops to 0.68 with split transmitters. With large receivers (Fig. 5-8), the effect of changing from 1:2 to 2:2 is seen as an increase in received power, as the maximum
Prx is increased from 1.82 to 9.45. Likewise, with 3 receivers, Fig. 5-9 demonstrates that there is also an increase in received power, while the maximum efficiency remains
about the same. Using the split transmitter decreases ηc to 0.67. It seems that using multiple transmitters that are spatially separated from each other reduces efficiency and received power as the fringing fields are dissipated into the nearby environment instead
62 Figure 5-10. Total received power as a function of RL, and its 95% confidence intervals with 5% component tolerance (red lines), 10% component tolerances (blue dashes), and 20% component tolerances (black dash-dot), for both 1:2 (left) and 2:2 (right) cases.
of coupling into a neighboring coil. Table 5-2 gives the maximum Prx and ηc for each test.
To investigate the sensitivity to component variation, a Monte Carlo simulation was run, assuming the components are normally distributed, with means given by the derived component formulas and with standard deviations, σ, such that 3σ is the component tolerance. These simulations were carried out at tolerance levels of 5%, 10%, and 20%, for the 1:2 and 2:2 configurations, using the large receivers. One receiver was fixed at 500 Ω and the other was swept from 60 to 4000 Ω. Figure 5-10 shows the 95% confidence intervals for total received power at the three tolerance levels. Figure 5-11 shows the 95% confidence intervals for total efficiency at the three tolerance levels. As can be seen, the power is skewed low, and with tighter tolerances for 1:2 than for 2:2. Efficiency is skewed high, with tighter tolerances for the 2:2 system than for the 2:1. This skew low in the power confidence intervals and skew high in the
63 Figure 5-11. Total efficiency as a function of RL, and its 95% confidence intervals with 5% component tolerance (red lines), 10% component tolerances (blue dashes), and 20% component tolerances (black dash-dot), for both 1:2 (left) and 2:2 (right) cases.
efficiency confidence intervals shows that the system is not optimized for maximum power delivery but rather efficiency. This makes sense, as all of the component selection for the system is done on the basis of efficient operation of the class E. The 2:2 system’s
efficiency is less sensitive to component variation primarily because of Cout which
governs the phase range seen by the class E and thus its efficiency. Cout is larger in the 2:2 system, therefore its reactance is less sensitive to variations. For total received power, the 1:2 system is less sensitive than the 2:2 system to component variations, because the two receivers in the 2:2 system can vary more independently due to the decoupling effect. 5.4 Conclusion
Inductive wireless power transfer between M primary coils coupled to N secondary coils is derived analytically and demonstrated experimentally for M = 1, 2 and N =
64 1, 2, 3. Using multiple primary coils in parallel has advantages over a single primary coil. First, the reduced inductance of the transmitting coils makes the amplifier less sensitive to component variations. Second, with multiple receiving coils, the power delivery to an individual receiver is less sensitive to changes in the loads attached to other coils, decoupling receivers from each other. In addition, using multiple transmitters is shown to increase received power with limited impact on coupling efficiency. The multiple transmitting coil architecture increases the feasibility and effectiveness of simultaneous multiple device charging as well as making the amplifier more robust to component variation.
65 CHAPTER 6 OPTIMAL PRIMARY COIL DESIGN FOR MULTIPLE COILS
6.1 Introduction
Chapter 4 presented a technique for coil design that ensures an even field.
However, to maintain even fields over greater areas requires a coil of higher inductance. From Chapters 2 and 5, this leads to increased sensitivity to component variation.
Chapter 5 develops the theory for multiple transmitting coils, which, among other things, reduces sensitivity to component variations. Naturally, the next step is to combine the multiple coil idea with the coil design technique. This chapter presents a coil design technique for multiple transmitting coils. Specifically, a system with two transmitting coils in parallel is designed. 6.2 Coil Design
The coil design for two transmitters differs from the system in Chapter 5 in that the dual-coil system there used two identical coils designed to work individually. Here, they are considered to be working together to establish a larger area of even fields. The basic principle behind design for multiple coils is the same as for single coils. That is, the geometry should be parameterized, and then the parameters can be optimized to give minimum field variations as measured by the coefficient of variation. Following the same base design from Chapter 4, where successive turns’ widths are related to the overall width by f :
f = 1 − (1 − (N − n + 1)/N)k (6–1)
and with the corners blunted by a fraction δ. Consider a two-coil system with overall y dimension W and x dimension L. The two coils overlap in the y direction by some amount b. A list of coordinates (x, y) is generated by the method in Chapter 4 for the coil associated with parameters (N, k, ∆) with dimensions L and w:
66 Table 6-1. Design parameters. N 15 k 2.64 ∆ 0.10 a 0.40 b 9.00 cm W 35.00 cm
w = (W − 2b)/2 (6–2)
The y coordinates are then skewed by an exponent a and offset by b:
a y1 = w(y/w) + b (6–3)
a y2 = −w(y/w) − b (6–4)
This stretches out the coils, so that the turns are spaced further apart in the region where the two coils overlap. So essentially, the multiple coil design is the same as single coil design with two additional parameters, skew a and overlap b.
For a two coil system with W =35cm and L=25cm, the optimum parameters are given Table 6-1. These dimensions were chosen such that the coil could accommodate a laptop equipped with a receiving coil.
Fig. 6-1 shows the corresponding coils and Fig. 6-2 shows the MQS estimate of the
field distribution. 6.3 System
The dual coil system was tested, using the parameters in Table 6-1 and a receiver of 8cm by 10cm with 6 turns. The component values for the class E were selected according to the general principles described in Chapter 2 by sweeping the component values and hand tuning. The values used are given in Table 6-2. Figs. 6-3 and 6-4 show the coils and circuits.
67 Figure 6-1. Coil layout.
Figure 6-2. Calculated z-directed magnetic field, assuming 1 A current (A/m).
68 Figure 6-3. Transmitter test setup.
Figure 6-4. Overlap of dual transmitter coils.
Table 6-2. Component values.
Crx 53.80 nF Cout 13.50 nF Lout 9.12 µH Ct 18.00 nF
69 Table 6-3. Summary of system performance. Size 25 cm by 35 cm Peak delivered power 5.62 W Peak total efficiency 59.3% Peak coupling efficiency 75.6% COV 17.8% Coil 1 self-inductance 50.41 µH Coil 1 resistance 0.44 Ω Coil 2 self-inductance 51.27 µH Coil 2 resistance 0.44 Ω Coil 1 and 2 mutual inductance 4.87 µH Total self-inductance 27.86 µH Total resistance 0.22 Ω
6.4 Testing
To test the system, the received power, transmitted power, and input power, were
measured at a load resistance of 100 Ω at 5cm grid points over the coils to evaluate the spatial variability. In addition, the received power, transmitted power, and input power, were measured at loads of 75, 100, 250, 500, 750, 1000, and 4000 Ω and the receiver centered on the transmitter. 6.5 Results
Fig. 6-5 is a plot of the received power over the area of the coils. This shows the spatia1 uniformity of the power distribution over the area of the coil (−17.5 < y < 17.5 and −12.5 < x < 12.5). There is more variability in the y direction than in the x direction.
The coefficient of variation is 17.79%. The ratio of receiver area to transmitter area is about 0.09, whereas is Chapter 2 is was 0.12, which explains the greater variability.
Fig. 6-6 shows the plot of the load resistance response. The response trend is
similar to those seen for systems in Chapters 2, 4, and 5. In general, the efficiencies are
lower than those presented in Chapter 4 due to the lower mutual inductance between
the receiver and transmitter. Table 6-3 gives a summary of the performance characteristics of the system.
70 Figure 6-5. Received power (W) as a function of the location of the center of the receiving coil.
Figure 6-6. Power (W) and efficiency (%) at loads from 75 Ω to 4 kΩ.
71 6.6 Conclusion
This chapter has demonstrated the application of the design strategy for a single coil to multiple coils. This confers the advantages of large coil size and reduced primary inductance. The primary challenge in designing such a coil is achieving an even power delivery regardless of receiver position, in order to accommodate multiple devices. Such a coil design was achieved through optimization, and the 25 cm by 35 cm coil was built and tested with a class E power amplifier and a 8 cm by 10 cm receiving coil. It was found to have a maximum efficiency of 59.3% and a maximum power delivery of 5.62 W. At a fixed load, the power delivery has a coefficient of variation of 17.78% as the receiving coil’s position is varied on the transmitter.
72 CHAPTER 7 INCLUSION OF FERRITES
7.1 Introduction
In a near-field wireless power system, if the device under charge is electrically conductive, or if there are electrically conductive objects underneath the transmitting coil, the fields generated by the primary will be dissipated instead of contributing to power transfer. It would be desireable to have an EM shield that would allow or even enhance power transfer without letting fields dissipate. For instance, a cell phone equipped with a receiving coil would need a shield between the coil and the battery in order for there to be effective wireless power transfer.
One way to do this is through use of high magnetic permeability materials, such as ferrite, backed by copper ([35, 36]). The coil is placed on top of ferrite, which is on top of the copper. The principle of operation is based on the magnetic field boundary conditions on the normal and tangential components of the field:
Ht1 = Ht2 (7–1)
Bn1 = Bn2 (7–2)
µ0Hn1 = µr µ0Hn2 (7–3)
where subscript t denotes the tangential component, n denotes the normal component, material 1 is air, and material 2 is ferrite. Since the tangential field is
Figure 7-1. Diagram of ferrite shielding.
73 continuous, and the normal field drops in magnitude due to the large increase in permeability across the boundary, the field is guided through the ferrite and reduced in magnitude. The fields reaching the copper are thus reduced so little power is dissipated in the copper. So instead of uselessly generating currents in conductive objects behind the coil, the ferrite guides the field through the coil. In terms of the wireless power circuit parameters, the inductance is increased and the parasitic resistance is decreased.
This chapter presents several aspects of the use of ferrite for shielding. Sections 7.2 and 7.3 discuss theoretical effects of ferrite properties on the coil inductance and losses.
Section 7.4 uses numerical simulation to establish width and thickness effects of a ferrite shield on the inductance and resistance. Finally, Section 7.5, presents experimental evaluation of several commercial ferrites.
7.2 Inductance Estimation
[37] presents a derivation of the effect of a semi-infinite, lossless ferrite substrate on the inductance of a radially symmetric coil using image theory. The end result is
2µ L = L (7–4) µ + 1 0
where L0 is the free-space inductance of the coil. So, in the limit, the ferrite inductance is twice the free-space inductance. Clearly this sort of impact on the inductance matrix will require careful consideration of circuit tuning. For ferrites of finite thickness, but infinite planar extent, the effects on the inductance of radial coils is explored analytically in [38–40]. However, the real case of a finite ferrite shield
over a finite copper shield is more complicated. This could potentially be handled by Schwarz-Christoffel mapping [41]. By mapping the semi-infinite solution for the
vector potential A to the finite geometry the inductance in the finite geometry could be estimated. If the vector potential in the semi-infinite geometry is denoted A(z), and
the Schwarz-Christoffel transform between the original coordinates z and the new coordinates w is w = g(z), the mapped vector potential is A¯:
74 A(g−1(w) A¯(w) = (7–5) g0(g−1(w)) and the new inductance is
H Ads¯ L = C (7–6) I where C is the path of the coil. A simple way to evaluate the effect of a ferrite shield on inductance would be to use an empirical function of the coil properties and the ferrite properties to calculate an effective permeability, similar to the effective dielectric constant used in microstrip design [42]. Defining this constant as
L µeff = (7–7) L0
where L is the inductance with ferrite, and L0 is the inductance in free space. A simple geometrical parameter for describing the coil is the coil’s length l. The ferrite can be described by its relative permeability µr and by its thickness h. Using a functional relationship functional relationship similar to that of the effective dielectric,
2µr µeff = C (7–8) µr + 1 h C = A(1 + B )n (7–9) l
where A, B, and n are fitting parameters. To determine the values to use for these parameters, the coil inductance of circular coils of varying diameter and number of turns was measured in free space and over three different ferrites, of µr 125, 2000, and 2100. Using these measurements, the best-fit A is 0.8128, B is 97.7237, and n is -0.09.
75 Figure 7-2. Empirical µeff predictions (red x) and observations (blue circle).
Figure 7-3. Flux-field hyteresis loop
Fig. 7-2 shows the measured data and values estimated by the best fit function. Of course, this function is a gross simplification, as it ignores many geometrical effects, but it provides a quick estimate of a particular ferrite’s effect.
76 7.3 Loss Estimation
Besides the effect on the inductance matrix, a ferrite shield introduces losses due to two effects. The first is hysteresis losses from the alternating fields. Due to saturation, the peak flux and field is
0 B = min(Bs , µ Hp) (7–10) B H = H + (7–11) c µ0
Using these, the hysteresis loss can be estimated one way, approximating the B − H curve as a parallelogram (Fig. 7-3)
Physt = fHc B (7–12)
where Hc is the coercivity. Or, using the complex permeability µ0 + jµ00,
00 2 Physt = πf µ H (7–13)
Conductive losses are the second loss mechanism and are usually less than the
hysteresis losses. These can be estimated according to [43] as:
(πhfB)2 P = (7–14) cond 6ρ where ρ is resistivity.
7.4 Thickness and Width Effects
A practical consideration for ferrite shielding is the necessary dimension and
necessary permeability of the shield material. To determine what size and permeability shield would be sufficient, finite-element MQS simulations were run in Ansoft Maxwell of a square coil resting on a ferrite of thickness h, permeability µ, and relative width F . F is defined as the ratio of the ferrite width to the coil width. These three variables were
swept and at each (h, µ, F ) point the ratio of the inductance to the free-space inductance
77 Figure 7-4. Effects of thickness and relative width of ferrite on inductance.
was calculated and plotted in Fig. 7-4. This ratio shows the expected increasing trend
with increasing h, µ, and F . This alone is insufficient to determine a requirement on shield size.
The losses must also be considered. Fig. 7-5 shows the ratio of the resistance to
free space resistance. As F increases, this ratio decreases, approaching 1 because
fewer fringing fields are being dissipated in the copper backing. The effect of F flattens off around 1.2 so the shield width should be about 1.2 times the coil size. More concretely, the 1/e calculated folding length for the average of the curves in Fig.
7-4 is 1.08 and in Fig. 7-5 is 1.09. Using 1.2 as a guide provides some degree of safety
margin for design purposes.
For µ =600, a practical value, the effect of h variation is small, so it should be possible to use a shield as thin as 0.2 mm or 0.3 mm.
78 Figure 7-5. Effects of thickness and relative width of ferrite on resistance.
7.5 Experimental Evaluation
To evaluate the losses and shielding effectiveness of potential commercially available ferrites, current was run through a solenoid test coil placed over the ferrite or the ferrite/copper combination. The current and voltage waveforms were captured for one period to calculate the input impedance as follows:
I = Ipcos(ωt) (7–15)
V = Vpcos(ωt + φ) (7–16) V |Z| = p (7–17) Ip φ = arccos(IV¯ ) (7–18)
The power dissipated is:
79 Table 7-1. Ferrite properties. 0 00 Ferrite µ µ Bs (T) Hc (A/m) ρ (Ωm) FairRite 42 2530 118 0.40 5 5×104 3M 600 710 125 0.273 68 108 Ferrox 3C96 2000 10 0.50 13 5
1 P = I 2<(Z) (7–19) 2 p For a solenoid coil, the field strength produced is,
NIp Hp = (7–20) hcoil
where N =18 turns and hcoil , the height of the solenoid, is 20 mm with a diameter of about 20 mm for the particular test coil. The test results for several ferrites and thicknesses with and without copper shielding are presented in Table 7-2.
The resistance (R), reactance (X ), power dissipation (P), and inductance (L) were calculated using the voltage and current waveforms. L0 and R0 are the free-space inductance and resistance of the solenoid test coil. The most desireable material would have an R/R0 as close to 1.00 and a L/L0 greater than 1.00, for as thin as possible ferrite, over a copper backing.
Samples of FairRite 42 were only available in one thickness, 1.00 mm. It shows similar performance with the copper backing and without. This indicates it is an effective shielding material. However, 1.00 mm may be too thick and heavy to be used as a shield in a portable device. 3M 600 samples were obtained at three thicknesses, 0.4 mm, 0.3 mm, and 0.2mm.
00 It has greater hysteresis losses than the FairRite 42, as its µ and Hc are higher. Its conductive losses are lower, with its greater resistivity. From the test data, the 0.3 mm thick 3M 600 should provide adequate shielding.
80 Table 7-2. Ferrite experimental evaluation with solenoid coil.
Ferrite h (mm) Ip (A) Hp (A/m) P (W) X (Ω) R (Ω) L/L0 R/R0 No ferrite - 2.88 2592 0.51 13.19 0.11 1.00 1.00 FairRite 42 1.00 2.80 2520 0.46 15.71 0.12 1.19 1.09 FairRite 42 w/ Cu 1.00 2.84 2556 0.49 15.77 0.12 1.20 1.09 3M 600 0.40 2.64 2376 0.52 15.76 0.15 1.19 1.36 3M 600 w/ Cu 0.40 3.16 2844 1.01 14.90 0.16 1.13 1.45
81 3M 600 0.30 2.72 2448 0.45 15.88 0.12 1.20 1.09 3M 600 w/ Cu 0.30 2.32 2088 0.39 15.00 0.14 1.14 1.27 3M 600 0.20 2.72 2484 0.41 15.94 0.11 1.21 1.00 3M 600 w/ Cu 0.20 3.24 2916 0.97 15.06 0.19 1.14 1.73 Ferrox 3C96 4.76 2.56 2304 0.48 15.94 0.15 1.21 1.36 Ferrox 3C96 w/ Cu 4.76 2.40 2160 0.40 15.17 0.14 1.15 1.27 Ferrox 3C96 3.18 2.84 2556 0.58 16.06 0.14 1.22 1.27 Ferrox 3C96 w/ Cu 3.18 3.20 2880 1.09 15.25 0.21 1.16 1.91 Ferrox 3C96 1.59 3.04 2736 0.68 15.00 0.15 1.14 1.36 Ferrox 3C96 w/ Cu 1.59 3.08 2772 0.59 13.68 0.25 1.04 2.27 Of the three test ferrites, the Ferrox 3C96 has physical properties indicating the lowest hysteresis and highest conductive losses. In terms of L/L0, and R/R0, it is the worst performing at the thicknesses available as samples. Of course, thicker ferrites, all other things being equal, should have greater power dissipation due to the greater volume of material. In addition, the copper backing has the effect of lowering L/L0 and raising R/R0 but its effect on power dissipation in tests appears ferrite and thickness dependent. Overall, the best practical shielding would be the 3M 600 at a thickness of 0.3 mm.
7.6 Conclusion
This chapter presented several aspects of the use of ferrite for shielding: theoretical effects of ferrite properties on the coil inductance and losses, numerical simulation to establish width and thickness effects of a ferrite shield on the coil electrical properties, and empirical formulas to estimate these effects. In addition, the chapter presented experimental evaluation of several commercial ferrites. 3M ferrite of µ =710 and thickness 0.3 mm and 20% wider than the coil would be a good shield for wireless power applications.
82 CHAPTER 8 BAYESIAN LOAD/FAULT TRACKING
8.1 Introduction
A wireless power system could charge multiple devices of different types, simultaneously.
Of course, one concern is metal objects near or on the transmitter. These effectively short out the transmitted fields. Another concern is devices under charge “fighting” each other for power as described in Chapter 5, which could lead to instabilities or perhaps excessive charge times. Naturally, then, it would be desirable to detect the presence of faults, the number of loads, and their battery charge status or some similar measure.
The problem of load or fault detection is essentially a problem of state estimation
[44]. The state estimation problem posed in this chapter is a combination of discrete and continuous states. The problem of detection and estimation in the wireless system is compounded by the fact that only two measurements are available on the transmitter: DC input current and voltage on the transmitter coil. There are multiple possible states that are a combination of continuous and discrete random variables: different numbers of loads and their load currents or charges, or possible fault conditions. In addition, when multiple loads are present, they influence eachothers’ received power. So, a detection/estimation scheme for this system should be able to handle discrete and continuous variables. It should also be robust, because a failed detection of a fault condition could be dangerous.
There are many qualitative or quantititative methods for this kind of estimation in industrial processes and other complex systems [45–47]. Three families of techniques are outlined below, all of which are some form of Bayesian tracking.
The Kalman filter is a powerful tool for estimating the state of a process in the presence of process and measurement noise. The basic idea is to use the known system and measurement dynamics and a time series of measurements to estimate the current state of the system [48]. It is used for continuous variables. At each time
83 step there is a prediction and update step. The prediction step uses the known system dynamics and the previous estimate of the state to estimate the current state. In the update step, a Kalman gain is applied which essentially weights the estimate and the measurement to update the current estimate. The classic Kalman Filter has requirements of linearity and normality but these can be relaxed somewhat in alternate versions, such as the Extended Kalman Filter and the Unscented Kalman Filter.
Hidden Markov Models are useful in state estimation when states are from a discrete set. An Hidden Markov Model consists of a set of finite states, transition probabilities between theses states, and probability distributions of observation symbols conditioned on these states [49]. The Hidden Markov Model problem is, given a sequence of observation symbols, what is the sequence of states that caused these symbols? The famous Viterbi algorithm is one approach. The important restriction is that the state space is discrete.
Particle filters are a flexible method of state estimation using Markov Chain Monte Carlo techniques. The essential idea behind a particle filter is to use random samples to represent the probability distribution of possible states, updating the samples as the system evolves with time and with new measurements [50]. It combines elements of the Kalman Filter and Hidden Markov Model. The fact that it uses samples rather than a distribution lifts any restriction on normality or even linearity. In addition, it can be extended to include combinations of discrete and continuous states, hierarchies of states, and risk-weighted states [51, 52]. This means that improbable but dangerous states will not be missed. Another key feature is its computational simplicity, making it easier to implement in a less powerful microcontroller as might be used in a process monitoring situation. For these reasons, this chapter will focus on the particle filter in a wireless power system.
84 8.2 Technology/Data
In this chapter, the transmitting coil is a 16 by 18 cm coil with 13 turns and the receivers are all 4 by 5 cm coils with 6 turns. The transmitting coil is designed according to the technique described in Chapter 4.
Rather than use test data, this chapter uses simulations of the system with a model already developed in Matlab, which makes use of classic analytical solutions for the class E amplifier [17, 19] and numerical integration for the calculation of coil inductances and mutual inductances [24]. In the system, there is measurement noise (thermal) and process noise (due to load fluctuations). Components are assumed to have zero tolerance and the receiving coils’ positions are assumed fixed once placed on the transmitting coil.
8.3 Theory/Methods
8.3.1 State/Measurement Model
Five possible discrete states are considered: zero through three loads, or fault mode (metal object on the transmitter). For no load or fault mode, there is only the discrete state to be estimated. For the 1-3 load cases, the discrete state (number of loads) as well as the continuous state (charge status of the loads’ batteries) must be estimated. The governing equation is for the charge in the battery and its time rate of change, where vectors are used to indicate the possibility of a multiplicity of loads. Given the DC received power (PDC ), and the fixed regulator output voltage (Vreg),
∂Q~ P~ = ~I = DC (8–1) ∂t Vreg
Relating charge to equivalent resistance, then discretizing:
∂f (R~ ) P~ = DC (8–2) ∂t Vreg ∂f ∂R~ P~ = DC (8–3) ∂R ∂t Vreg
85 µ ¶ −1 ∂f P~ DC (R~ k−1) R~ k = R~ k−1 + + nk−1 (8–4) ∂R Vreg where k is time index and n is process noise. The measurement equation is
(Vin, IDC ) = h(R~ k ) + νk (8–5)
where h is the measurement function and ν is measurement noise. The functional
relationships P~ DC (R~ k−1) and h(R~ k ) are known from derivations in previous chapters and already coded in Matlab. It suffices to say they are nonlinear. f relates charge to load
resistance and is defined as follows for the purposes of this chapter:
R Q = Q0 (8–6) R0
for R ≤ R0
R − R0 Q = (Q1 − Q0) + Q0 (8–7) R1 − R0
for R0 < R ≤ R1
Q = Q1 (8–8)
for R > R1. Q, Q1, and Q0 are in Coulombs. R, R0, and R1 are in Ω. For this
instance, Q0 = 36, Q1 = 3600, R0 = 1, and R1 = 100. 8.3.2 Particle Filter Algorithm
The particle filter algorithm is composed of several steps, detailed in the following
sections. 8.3.2.1 Dataset generation
This first step for load tracking is not technically a particle filter step but is necessary
for tests. A ”truth” dataset is generated using the Monte Carlo Markov Chain method and state/measurement model previously described. This provides the ”measurements”
86 Vin and IDC to be used and mode (mk ) state values (Rk ) to be estimated. This chapter uses a one-hour sequence of measurements with samples every one second. 8.3.2.2 Initialization
The first step is to generate an initial group of N particles. This is done with a
i uniform prior distribution for the discrete modes mˆ 0 and Gaussian distribution for each of ˆ i the loads R0 (i is particle index). At the end of this step, each particle has been assigned a discrete mode, and if the mode has loads asociated with it, the loads have resistance values. 8.3.2.3 State
The next step is to update the particles according to the state model. The Markov transition matrix aij is used at this point to determine if any particle will move to a new mode. A (0,1) uniform random variable u is generated and the CDF for each current mode is calculated (the current mode is i):
j=n Fn = Σj=1aij (8–9)
This is used to determine the next mode. For each u, when
Fn−1 < u < Fn (8–10)
the next mode is mode n. If there is a mode change from time k − 1 to time k, the states of any loads are re-initialized. For example, if the transition is from no-load to two-load, the two loads are assigned initial state values as described in Section 8.3.2.2.
8.3.2.4 Measurement
Using the measurement equations, estimates of Vˆin and ˆIDC are obtained. Then, using the (assumed known) distributions of measurement noise, and measured Vin
i and IDC , the conditional probabilities of each particle i at time k (wk ) are obtained. The
i collection of N wk ’s are then scaled so that they sum to one.
87 i i | ˆ ˆ wk = wk−1p(Vin, IDC Vin, IDC ) (8–11) ³ ´ i − ˆ − ˆ |~ = wk−1Pν (Vin Vin, IDC IDC ) Θν (8–12) i i wk wk = i (8–13) Σi wk ~ where Pν is the noise pdf and Θ is its parameter vector. 8.3.2.5 Update
These w are used to update the particles. Two methods are considered in this chapter. The first is as described above, keeping the same particles but rescaling their
j weights. The second incorporates resampling the particles using the CDF Ck from their weights
j i=j i Ck = Σi=1wk (8–14)
This is used to resample by generating N instances a (0,1) uniform random variable u. For each ui , when
j−1 i j Ck < u < Ck (8–15)
the new resampled particle i is the old particle j, and all the resampled particles weights are considered uniform. This is supposed to allieviate problems of the distribution degenerating to a single particle with weight 1.
These methods will be referred to as particle filter with and without resampling.
8.3.2.6 Estimate
The estimate at each time step is the weighted (by w) sum of the particles’ values.
88 i i mˆ k = Σwk mk (8–16)
ˆ i i Qk = Σwk Qk (8–17)
i i = Σwk f (Rk ) (8–18)
The modes are given numerical values as follows: m = 1 is one load, 2 is two loads,
3 is three loads, 4 is no load, and 5 is fault mode. 8.3.3 Tests
The state model, measurement model, and particle filters (with and without resampling) described above were used with a generated data set. N was set to be
10, 100, or 1000. The prior distribution of modes was assumed uniform (p = 1/5 for each mode), and transition probabilities between modes were defined aij = 0.996 if i = j, and 0.001 otherwise. Process noise and measurement noise were assumed Gaussian,
though it should be noted that this is not a requirement of the particle filter technique. Noise can come from any arbitrary distribution with a particle filter. The standard deviation and mean of the prior distribution of the load resistances (for initialization) are 0.1 and 1 Ω. The standard deviation and mean of the process noise are 0.1 and 0.0 Ω.
For Vin, the mean of the measurement noise is 0.0 V and the standard deviation is 0.1 V;
for IDC , the mean of the measurement noise is 0.0 A and the standard deviation is 0.01 A. 8.3.4 Implementation
The particle filter load tracking scheme was implemented in a physical system,
using the test setup as in Chapter 5, with the 4 cm by 5 cm receivers. The component
selection is the same. The MATLAB particle filter code was implemented in C++ and combined with the code used for controlling the electronic loads, DC source,
oscilloscope, and function generator. The Vin and IDC measurements were obtained from the oscilloscope and DC source, and the electronic loads were programmed to behave
89 according to the piecewise-linear model established earlier. Process noise was 0.1 Ω; voltage noise was 0.5 V; and current noise was 0.03 A. The transition probabilities between modes were defined aij = 0.9 if i = j, and 0.025 otherwise. The system was tested in different modes individually, and in a sequence of no load, 1 load, 2 loads, 3 loads, 2 loads, 1 load, 0 load, and fault. For the sequence tests, the particle filter used a hierarchical implementation, in which first the no load and fault mode are ruled out. If Idc was greater than 1 A then fault mode was determined. If Idc was less than 0.4 A, and Vin was greater than 70 V, the system was determined to be in 0 load condition. Otherwise, the particle filter was as in the MATLAB simulations.
8.4 Simulation Results
Figure 8-1. Generated measurements in (Vin,IDC ) space.
Fig. 8-1 shows the generated “truth” dataset in the (Vin, IDC ) space. The points are color coded to indicate the discrete mode. Notice how close the different modes
90 are in the feature space. Actually, in real wireless power systems tested in the lab, this separation is greater, so discrimination should be easier.
Figure 8-2. True (blue) and estimated (red) mode for N=10, with resampling.
Figs. 8-2, 8-3, and 8-4 show the estimated and true mode time series (mˆ k and mk ) for N as 10, 100, and 1000. What is the physical interpretation of the “truth” time series?
The system starts in fault mode (say a metal sheet is placed on the transmitting coil), then a load is placed on the transmitting coil. This load charges, then the system goes into fault mode again. The metal sheet is removed and the system is in no load state. One load is placed on the transmitting coil; it charges, then another load is placed; it charges, then it’s removed. The system is somehow placed in fault mode again, followed by a long stretch of time with three loads. This is followed by a brief unloading, then one load, then two loads, then a short unloading, then fault mode. Finally, two loads are placed on the transmitting coil to charge.
91 Figure 8-3. True (blue) and estimated (red) mode for N=100, with resampling.
The estimated mode matches the true mode more closely as N is increased.
Transitions between modes are missed because none of the particles, after resampling, are in the new mode, so it is impossible to detect that change. Essentially, with resampling, the particle’s distribution lacks enough variability to detect sudden changes.
Figs. 8-5,8-6, and 8-7 show the estimated and true mode time series for N as 10,
100, and 1000. In general, without resampling, the estimate is more variable because all of the particles are there, instead of being resampled. On one hand, this is bad, because the estimate is ”noisy”. On the other hand, this is good because the increased variablity allows the filter to capture transitions that the filter with resampling misses.
Plots of the charge state time series are not included as they change not just in value but also in dimension (1, 2, or 3 loads) over time. Fig. 8-8 shows the mode and state root mean square error (RMSE) as a function of N, for both with and without resampling. State RMSE is calculated in terms of the
92 Figure 8-4. True (blue) and estimated (red) mode for N=1000, with resampling. charge status of the loads, not the load resistances. RMSEs decrease with increasing N as would be expected. The mode RMSEs are lower for the particle filter without resampling than with resampling; the state RMSEs are lower with resampling than without. With resampling, the particles have finer granularity (lower variance in mode and charge; see Fig. 8-9); without resampling, the particles have coarser granularity
(greater variance in mode and charge; see Fig. 8-10). The implication of this is that the filter without resampling will be more able to catch the sudden transitions between modes, but not the gradual charging of the device or devices. With resampling, transitions are more likely to be missed because it could be that after resampling that all the particles have the same mode.
93 Figure 8-5. True (blue) and estimated (red) mode for N=10, without resampling.
8.5 Measured Results
Fig. 8-11 shows the predicted and observed (Vin,IDC ) space for all 5 modes, tested individually, with N=10000. From this it is evident that the state/measurement model is accurate. The primary difficulty is in detecting transitions between modes. Figs. 8-12 through 8-17 shows the predicted and observed mode time series, and
the predicted and observed state and measurement variables for N=100, N=1000, and N=10000. Due to the hierarchical technique, no load and fault (modes 0 and 4) are detected readily for all N values. However, results are generally poor.
For N=100 and N=1000 (Figs. 8-12 and 8-14) the estimated mode is usually within 2. For N=10000 (Fig. 8-16) the estimate is within 1. The poor accuracy could be due to the numerous uncertainties in the system that are unknown in the tracking model. For instance, the true transition probability matrix, process and measurement noise are all unknown. A more refined technique might not rely on such precise knowledge.
94 Figure 8-6. True (blue) and estimated (red) mode for N=100, without resampling.
The load resistance and power delivery estimates for N=100 and 1000 (Figs. 8-13
and 8-15) show poor performance, due in no small part to the incorrectly estimated mode. This performance improves for N=1000 (Fig. 8-17) as the mode is more closely
estimated. Without a higher N, the continuous and discrete state tracking accuracies are probably insufficient. This number of particles (more than 10000) is probably impractical for an on-board microprocessor. 8.6 Conclusion
In conclusion, the particle filter does work for fault detection/load tracking. However, the number of particles to achieve satisfactory results is impractical for an on-board microprocessor. Two changes could be implemented. One is to only do selective resampling: if the probability mass function of the particles has insufficient entropy (or some other metric) only then is resampling conducted. This would maintain variability while avoiding degeneracy of the distribution. The state model could be simplified
95 Figure 8-7. True (blue) and estimated (red) mode for N=1000, without resampling. which would reduce computational complexity for implementation on an ARM or similar microprocessor. In addition to changes for performance, model changes could be implemented.
The charge-resistance model used in this chapter was fairly arbitrary; a measured charge-resistance curve for a particular device should be used. Different receiver types and relative positions could be included. In general, the particle filter algorithm is an effective tool for challenging nonlinear discrimination problems such as wireless load tracking.
96 Figure 8-8. RMSE of mode and states.
Figure 8-9. Mode and charge estimate variance, with resampling.
97 Figure 8-10. Mode and charge estimate variance, without resampling.
Figure 8-11. Test of different modes in (Vin,IDC ) space, in real system.
98 Figure 8-12. True (blue) and estimated (red) mode for N=100, without resampling, in real system.
Figure 8-13. Predicted and observed power, resistance, and input voltage, and DC input current for N=100, without resampling, in real system.
99 Figure 8-14. True (blue) and estimated (red) mode for N=1000, without resampling, in real system.
Figure 8-15. Predicted and observed power, resistance, and input voltage, and DC input current for N=1000, without resampling, in real system.
100 Figure 8-16. True (blue) and estimated (red) mode for N=10000, without resampling, in real system.
Figure 8-17. Predicted and observed power, resistance, and input voltage, and DC input current for N=1000, without resampling, in real system.
101 CHAPTER 9 MIDRANGE WIRELESS POWER TRANSFER
9.1 Introduction
In previous chapters, the receiver-transmitter distance considered was less
than 5 mm. This is appropriate for the primary application considered, charging battery-operated electronics. However, for some applications it may be desirable to extend the receiver/transmitter distance to about the size of the coils themselves. This is considered midrange coupling. The approach taken in [7, 53] is to use high-Q, electromagnetically resonant structures to form a strong coupling. The frequencies used are in excess of 10 MHz and coupling efficiencies of 90% are achieved at distances of 75 cm. In addition, rather than the conventional inductive coupling equations considered earlier in this dissertation, [7] uses coupled mode theory [54–56]. [53] uses a higher frequency (10 MHz) and coupled mode theory, relying on the self-capacitance of the coils to achieve resonance, though total efficiency is low (∼40%), due in part to their selection of a Colpitts oscillator as the driving circuit. [57] uses this resonant technique but with lumped capacitors to power an LED at a distance of a few centimeters. The papers using resonance all rely on a total of four coils for every receiver-transmitter pair: a transmitting coil, two resonantly coupled intermediary coils, and a receiving coil. This chapter tries to extend the architecture considered in previous chapters to midrange distances while maintaining high total efficiency. First, the coupled mode theory analysis is compared to the inductive coupling analysis. Next, coil design is reconsidered for midrange. Then, design rules are developed for the series-parallel architecture (and others) to extend the class E’s utility to midrange coupling. Inductance, frequency, and circuit topology effects are tested on an actual system and evaluated in terms of power and efficiency. The ideal topology for midrange is found and tested with regards to its sensitivity to component tolerances and relative receiver-transmitter positioning. Ultimately, one system is designed with peak total efficiency of 69.2% and peak received
102 power of 0.94 W and at 25 cm, and another is designed with peak total efficiency of 57.9% and peak received power of 3.78 W at 1 m.
9.2 Analysis
Coupled mode theory for two resonant objects can be described [53] by
da 1 = jω a − Γ a + jκa (9–1) dt 1 1 1 1 2 da 2 = jω a − Γ a + jκa (9–2) dt 2 2 2 2 1
where ai is a state variable of object i, such that its square has units of energy; ωi is the resonant frequency; Γi is the loss, with units of frequency; and κ is the coupling coefficient, with units of frequency.
To compare this to the inductive system, state variables, losses, and coupling are defined as follows:
r L a = i I (9–3) i 2 i Rpi Γi = (9–4) Li ωM κ = √ (9–5) L1L2 = ωk (9–6)
ωi Li Qi = (9–7) Rpi 1 ωi = √ (9–8) Li Ci
where Li s the coil inductance, Ii is the coil current, Rpi is the parasitic resistance, M is
the mutual inductance, Qi is the quality factor, and Ci is the resonant capacitance value. Strong coupling, necessary for midrange transfer, occurs when
κ √ À 1 (9–9) Γ1Γ2
103 which is equivalent to ωM p À 1 (9–10) Rp1Rp2
9.2.1 Coil Design
Coil design for the midrange system has different considerations than for the
near-field system. Whereas before, the primary concern was even field distribution, now the objective is maximal efficiency at a large separation distance. To do this, there must be strong coupling as described above, and the Q must be as high as
possible. This means the inductance should be as high as possible with the minimum parasitic resistance. So, both receiver and transmitter coils should have the maximum inductance for a given length of coil. Since we are considering a separation distance (d) approximately equal to the dimension of the coil (D), D is constrained by the desired
distance. Using the inductance formulas from [58], and for simplicity, assuming DC
resistance, the Q of a planar circular coil is:
ωL Q = (9–11) R (D/2)2N2 L = (9–12) 8(D/2) + 11N(2a) πDNρ R = (9–13) 2πa2 D2N2 L = (9–14) 16D + 88Na DNρ R = (9–15) 2a2 2ωa2 DN Q = (9–16) ρ 16D + 88Na
where ρ is resistivity, N is number of turns, and a is wire radius. For a square coil,
104 ωL Q = (9–17) R 2D L = N2 µ(ln(D/a) − 0.77401) (9–18) π 4DNρ R = (9–19) πa2 2ωNµa2 Q = (ln(D/a) − 0.77401) (9–20) ρ
Since D is fixed, Q can be increased by increasing a or N. For a given wire radius,
for both coils, Q is an increasing function of N. Intuitively, inductance goes as N2 and
parasitic resistance goes as N, there should be no “optimal” N. The constraint then
becomes the impact of the coil inductance on component selection, ie, the capacitor sensitivity discussed in Chapter 5. 9.2.2 Component Selection
In this chapter, the receiver and transmitter coils are considered indentical in geometry and thus inductance to simplify component selection. They are made resonant through proper capacitor selection:
L1 = L2 (9–21)
C1 = C2 (9–22)
C = (ω2L)−1 (9–23)
where C1 is the resonant transmitter capacitor and C2 is the receiver resonant capacitor. Because the capacitor selection is done for resonance, the design rules are different from those derived in Chapter 2. In addition, the desireable performance of the series-parallel topology for near-field induction (decreasing power delivery with increasing load) is not present when the receiver capacitor is chosen for resonance. Thus, three circuit topologies will be considered for the midrange system: the series-parallel,
105 Figure 9-1. Midrange class E series-parallel architecture. the series-series, and the T network. The following sections detail the design rules for each. 9.2.2.1 Series-parallel
Design rules for the series-parallel topology in resonance, shown in Fig. 9-1, are based on similar constraints to those developed in Chapter 2 but with the receiver capacitor chosen for resonance instead of for obtaining a specific R0 at the maximum real part of Zin.
4 2 2 3 2 Zin = ω M C RL − jω M C (9–24)
1 4 2 2 3 2 Ztx = + jωLout + ω M C RL − jω M C (9–25) jωCout
As before, Lout and Cout are chosen to meet Q and phase requirements, and Ct id chosen to obtain ZVS:
4 2 2 3 2 ωLout = Qω M C RL + ω M C (9–26)
3 2 −1 ωCout = (ωLout − ω M C(1 + tan(φ)ωCRL)) (9–27)
2 4 2 2 −1 ωCt = 2((1 + π /4)ω M C RL) (9–28)
where RL is the load resistance where the pahse is φ.
106 Figure 9-2. Midrange class E series-series architecture.
9.2.2.2 Series-series
Similarly, design rules for the series-series architecture, shown in Fig. 9-2, are
developed:
ω2M2 Zin = (9–29) RL 1 ω2M2 Ztx = + jωLout + (9–30) jωCout RL
Qω2M2 ωL = (9–31) out R ³ L ´ ω2M2 −1 ωC = ωL − tan(φ) (9–32) out out R ³ L ´ 2 2 −1 2 ω M ωCt = 2 (1 + π /4) (9–33) RL
9.2.2.3 T-network
The idea behind the T network, shown in Fig. 9-3, is to add an additional degree of
freedom to the design of the midrange series-parallel architecture so that a desireable
impedance response may be obtained. X2 and X3 are general reactances.
107 Figure 9-3. Midrange class E T network architecture.
4 2 2 3 2 Zin = ω M C RL − jω M C (9–34) ³ ´ 1 1 −1 Ztx = jX3 + + (9–35) jX2 Zin ω3M2CX + jω4M2R C 2X Z = jX + 2 L 2 tx 3 4 2 2 3 2 (9–36) ω M RLC + j(X2 − ω M C)
k 2R tan(φ)X 2 + kωMX (X − kωM) − k 4R2X X = L 2 2 2 L 2 (9–37) 3 − 2 4 2 (X2 kωM) + k RL
9.3 Preliminary Tests
To evaluate how to best maximize efficiency at midrange distances, the effects of
coil inductance, operating frequency, and circuit topology were investigated. In addition, the impact of the diode parasitic capacitance was investigated.
For tests in Sections 9.3.1, 9.3.2, and 9.3.3, the DC supply voltage was 6 V and the transistor used was the IRLR3410 NMOS. 9.3.1 Rectifying Diode Effects
The rectifying diode in the half-wave rectifier contributes some capacitance (10-30 pF). As the operating frequency increases, this becomes increasingly important in terms of achieving resonance. To demonstrate the diode’s effect, the system was configured with the series-parallel architecture at 761.79 kHz, with receiver and transmitter coils 30
108 Table 9-1. Component values. Component w/ diode compensation w/o diode compensation C2 0.614 nF 0.808 nF C1 0.808 nF 0.808 nF Cout 10.07 nF 10.07 nF Ct 6.63 nF 6.63 nF Lout 5.7 µH 5.7 µH
Figure 9-4. Diode effects on system performance.
cm square, with 8 turns, constructed of 420/42 Litz wire. The separation distance was 25 cm. The component selections for two systems are shown in Table 9-1; one system
has compensation for the diode capacitance, and the other does not. The values shown are measured values.
Fig. 9-4 shows the performance curves for the two systems. The non-compensated system has worse performance. Peak efficiency is about 20% lower, and the peak received power is about 0.05 W lower, compared to when the diode is taken into account.
109 Table 9-2. Component values. Component 240 kHz, 5 turns 757 kHz, 5 turns 761.79 kHz, 8 turns C2 18.45 nF 1.856 nF 0.614 nF C1 18.45 nF 1.858 nF 0.808 nF Cout 70.66 nF 10.06 nF 10.07 nF Ct 26.3 nF 10.01 nF 6.63 nF Lout 9.68 µH 5.7 µH 5.7 µH
Figure 9-5. Frequency and inductance effects on system performance.
9.3.2 Frequency and Inductance Effects
To evaluate the effects of coil inductance and frequency on power delivery, three systems were tested. A 240 kHz system, with 30 cm square coils of 5 turns; a 757 kHz system, with 30 cm coils of 5 turns; and a 761.79 kHz system, with 30 cm coils of 8 turns. The separation distance was 25 cm. Table 9-2 shows the relevant component values.
Fig. 9-5 shows the performance curves of the three systems. In general, as inductance and frequency go up (an increase in Q of the coils),the peak efficieny goes
110 Table 9-3. Component values. Component Series-parallel Series-series T-network C2 0.614 nF 0.808 nF 0.614 nF C1 0.808 nF 0.808 nF 0.808 nF Cout 10.07 nF 10.81 nF 8.96 nF Lp n/a n/a 1.05 µH Ct 6.63 nF 5.71 nF 11.1 nF Lout 5.7 µH 5.7 µH 5.7 µH
Figure 9-6. Topology effects on system performance.
up and peak power delivery goes down. This is because the real part of Zin increases with increasing M and ω. The higher real part is increasingly greater than the parasitics, leading to higher efficiency, and the higher real part results in lower current and thus lower power delivery. The strength of coupling as described by coupled mode theory mentioned in
Equation 9–10 is greater than 1 for only the 761.79 kHz system.
111 9.3.3 Topology Effects
Three topologies were evaluated: series-parallel, series-series, and the T-network,
all at 761.79 kHz with the 8 turn coils as before. The component selections are shown in Table 9-3.
Fig. 9-6 shows the performance curves for the three topologies. The series-series
has the highest power and efficiency, in addition to preserving the desireable trend of decreasing power with increasing power delivery. This is because for the resonant
series-series, Zin is a purely real, decreasing function of RL, and the phase of Ztx is monotonically increasing. For series-parallel and T-network, the impedance response is as in previous chapters, except the minimum phase is at a much larger load resistance.
Additionally, the topologies which have a parallel Crx have a higher voltage on the load, potentially beyond the voltage ratings of the rectifier and capacitor. , 9.3.4 Sensitivity
Having established the series-series as the best topology for midrange transfer, this section performs a sensitivity analysis with regards to: frequency, size, separation distance, and number of turns; and for the 761.79 kHz system described in the previous section the position and component tolerances.
à µ³√ ´ √ ¶ µN2 d 2 + D2 + D 2 d 2 + 2D2 + D M = D log √ π d d 2 + 2D2 − D ! √ √ + d + d 2 + D2 − 2 d 2 + 2D2 (9–38)
(9–39)
[59]
2V 2 R + R P = cc L p tx 2 2 2 (9–40) 1 + π /4 ω M + Rp(RL + Rp) 2V 2 R (R + R )2/k 2 + ω2M2 P = cc L L p rx 2 2 2 2 (9–41) 1 + π /4 (ω M + Rp(RL + Rp))
112 Figure 9-7. Effect of D, d, and f on total efficiency at N=8.
Figure 9-8. Effect of D, d, and f on received power at N=8.
113 Figure 9-9. Effect of N, f , and D on total efficiency where D = d.
Figure 9-10. Effect of N, f , and D on received power where D = d.
114 ηE = 1 (9–42) R (R + R )2/k 2 + ω2M2 = L L p ηc 2 2 (9–43) RL + Rp ω M + Rp(RL + Rp)
Using the series-series topology, the coil dimension (D), the separation distance
(d), the frequency (f ), and the number of turns (N), were swept simultaneously. For every D, d, f , N point, the maximum received power and total efficiency were calculated.
Fig. 9-7 shows the effects of D, d, and f on total efficiency at N=8. Efficiency is highest when d ≤ D and increases as f increases.
Fig. 9-8 shows the effects of D, d, and f on received power at N=8. Power is highest where d is slightly higher than D. This is because the weaker coupling leads to a smaller real part of Zin, leading to lower efficiency but higher power delivery. Fig. 9-9 shows the effects of N, f , and D on total efficiency where D = d. Efficiency increases with increasing f , increasing N, and increasing D. This is because the real part of Zin increases with f and mutual inductance, and mutual inductance increases with increasing number of turns and coil dimension.
Fig. 9-10 shows the effects of N, f , and D on received power where D = d.
Power delivery shows the trend of decreasing with increasing N while increasing with increasing f . As D increases, the highest power delivery occurs at lower N.
The systems’sensitivity to receiver placement was tested by measuring the load response at different vertical (z) and lateral (x, y) offsets. The performance curves are shown in Fig. 9-11. The offset vector for each curve is indicated in the legend. For example, [0, 7.5, 0] indicates the receiver’s center is displaced 7.5 cm in the y direction from the transmitter’s center. In general, the efficiency and power delivery decrease with increasing receiver offset. The worst performance is when the receiver is offset by [15, 15, 0] (the largest absolute offset): the received power is decreased by 0.5 W and the efficiency is decreased by 15%. However, these decreases are small enough for all
115 Figure 9-11. Coil offset effects on system performance.
other offsets (less than 0.2 W and 9%) to consider the system robust with regards to receiver/transmitter placement.
An identical Monte Carlo analysis as in Chapter 2 was performed for the series-series
system. The 95% confidence intervals of efficiency are shown in Fig. 9-12 and of power
are shown in Fig. 9-13. The efficiency confidence intervals exhibit a skew similar to that in Chapter 2, indicating the component selection is done to optimize efficiency. The power confidence intervals show high upper bounds and a peak around 100-200 Ω. This is because as the components vary, the system goes off-resonance. The off resonant system is like the system from previous chapters, with a peak power delivery rather then a monotonically decreasing power delivery. The power delivery can be higher (though the efficiency is lower) because at off-resonance the real part of Zin is smaller than at resonance.
116 Figure 9-12. Efficiency at nominal component values (black line) and 95% confidence intervals at 5% (red lines), 10% (blue dash), and 20% (black dash-dot) component tolerances for the midrange series-series system.
Table 9-4. Component values. Component Value C2 0.371 nF C1 0.372 nF Cout 19.3 nF Ct 3.02 nF Lout 5.7 µH
9.4 Synthesis 9.4.1 50 cm Separation
The system was configured with the series-series architecture at 758.1 kHz, with receiver and transmitter coils 50 cm square, with 8 turns, constructed of 100/40 Litz wire. The separation distance was 50 cm. The component selections for the system is shown in Table 9-4. The values shown are measured values.
117 Figure 9-13. Power at nominal component values (black line) and 95% confidence intervals at 5%(red lines), 10% (blue dash), and 20% (black dash-dot) component tolerances for the midrange series-series system.
Fig. 9-14 shows the performance curves. The peak efficiency is about 52.6% and the peak power delivery is about 0.13 W. Greater efficiency could be achieved with better Litz wire, as the coil parasitics for the 50 cm system were relatively high (2.6 Ω, compared to the 30 cm system’s 0.6 Ω). It should be possible to increase the total efficiency at greater distances by using less lossy coils, with Litz of higher gauge and greater strand number. 9.4.2 1 m Separation
A system with 1 m coil separation was constructed using 1725 strand, 48 AWG Litz wire to build coils of 1 m square and is shown in Fig. 9-15. The coils were attached to foam posterboard and hung from the ceiling. Number of turns, frequency, supply and gate voltages, and duty cycle were varied in an attempt to maximize efficiency. The
118 Table 9-5. Summary of 1 m tests. Maximum efficiency (%) 44.6 47.9 48.8 51.2 55.1 56.3 56.8 57.9 54.3 Maximum received power (W) 4.91 4.98 0.82 1.65 5.59 4.57 4.37 3.78 3.84 Turns 4 4 4 6 6 6 6 6 8 M (µH) 1.83 1.83 1.83 4.10 4.10 4.10 4.10 4.10 7.23 L1 (µH) 69.13 69.13 69.13 158.74 158.74 158.74 158.74 158.74 274.70 L2 (µH) 68.51 68.51 68.51 160.67 160.67 160.67 160.67 160.67 268.79 119 f (kHz) 730.00 706.00 701.00 510.34 510.34 511.88 511.88 513.50 438.55 Duty cycle (%) 50 50 50 50 50 50 45 40 50 Vds (V) 12 12 6 12 20 20 20 20 20 Vgs (V) 5 6 5 10 10 10 10 10 10 C2 (nF) 0.605 0.653 0.653 0.609 0.609 0.609 0.609 0.609 0.500 C1 (nF) 0.590 0.651 0.607 0.609 0.609 0.609 0.609 0.609 0.490 Cout (nF) 1.62 1.65 6.72 7.58 7.58 7.58 7.58 7.58 8.08 Ct (nF) 0.991 0.503 3.30 3.90 3.90 3.90 3.90 3.90 2.20 Lout (µH) 29.10 29.10 8.62 18.46 18.46 18.46 18.46 18.46 18.46 Figure 9-14. 50 cm system performance.
transistor used was the IRF640 NMOS, and the receiver rectifier was a full-bridge using IR10MQ060N diodes. Table 9-5 summarizes the results of many tests conducted to find the design
maximizing efficiency for the 1 m system. Increasing number of turns increases the mutual and self inductances, raising the Q and increasing efficiency; however, since the inductance is so high, the voltage on the transmitting coil’s leads is high enough to lead to arcing between adjacent turns. This makes tuning and testing the system for more than 6 turns difficult. Increasing frequency raises the real part of the impedance seen by the class E inverter, which would increase efficiency if not for the fact that the coil parasitics increase as well. Increasing supply (Vds) and gate voltages (Vgs) ensures the MOSFET switches completely into saturation when on. In addition, lower Vds is associated with stronger nonlinearity in the output capacitance of the MOSFET, making
120 Figure 9-15. 1 m system setup.
121 Figure 9-16. 1 m system performance.
tuning Ct for ZVS more difficult. Finally, changing the duty cycle can ensure ZVS and zero derivative switching (ZDS) at load resistances where otherwise this would not be the case.
Fig. 9-16 shows the performance curves of the best performing design. The peak efficiency is about 57.9% and the peak power delivery is about 3.78 W. The efficiency was improved by using less lossy Litz (for the 6 turn 1 m system at 513.50 kHz, the parasitics are about 2.6 Ω, while with 420 strand, 42 AWG Litz the parasitics were about twice as high) or by increasing the number of turns. 9.5 Conclusion
The near-field wireless power system considered in [16] and [60] was extended to midrange distances, where the coil size is comparable to the separation distance. The effects of coil inductances, frequency, circuit topology, and coil positions are tested on a system. It is found that the series-series topology is best for midrange power transfer;
122 that there is a tradeoff between power and efficiency, where efficiency increases and power decreases with increasing frequency and coil inductance; and that the resonant tuning makes the system robust with respect to variations in coil positions. Ultimately, one system is designed with peak total efficiency of 69.2% and peak received power of 0.94 W and at 25 cm, and another is designed with peak total efficiency of 57.9% and peak received power of 3.78 W at 1 m.
123 CHAPTER 10 FAR-FIELD WIRELESS POWER TRANSFER
10.1 Introduction
Figure 10-1. An example of a radiofrequency (RF) harvesting wireless sensor node [3].
Figure 10-2. An illustration of the solar power satellite (SPS) concept [61].
Far-field, or radiative, power transfer occours when the distance between the transmitter and the receiver exceeds the Rayleigh distance, D > 2d 2/λ, where d is the characteristic dimension of the transmitter, under the condition that d is larger than
λ. Two major applications of radiative wireless power transfer (WPT) are in ambient radiofrequency (RF) harvesting (Fig: 10-1) and the Solar Power Satellite (SPS) (Fig.
10-2). The idea behind the first technique is to convert the radio waves from communications into power. Since the power levels are low, typical applications include wireless sensors free from batteries and RFID tags equipped with small computational abilities
124 Figure 10-3. Atmosphere model schematic used in this chapter [63].
Figure 10-4. Canopy model schematic used in this chapter [64].
[3, 4].The SPS is an idea which came about in the late 1960s [5] and has seen received some recent revival [62]. The principle is to collect solar energy in space using a geosynchronous satellite with large solar panels and then convert the energy in microwave form and beam it to a receiving station on earth.
Both RF harvesting and SPS involve radiative transfer through participating media.
In RF harvesting, the participating media is absorbing and scattering vegetation and urban obstacles. In SPS, the media includes the ionosphere and the ice and water
125 particles in the atmosphere. This chapter will discuss the simplifying assumptions made and the resulting solutions to the radiative transfer equation as well as the practical implications, for two example frameworks: transmission through the atmosphere from space by a solar power satellite, and transmission through a vegetation canopy by an RF transmitting tower of some kind to an RF-harvesting sensor node. Fig. 10-3 illustrates the simplified example of the former considered in this chapter; Fig. 10-4
illustrates the simplified example of the latter. The framework for analyzing the power transfer is a finite plane-parallel absorbing and/or scattering medium, with an external beam of incident flux at the upper boundary. The coordinate system starts at 0 and extends downward. The lower boundary is reflecting. 10.2 Theory
Each of these frameworks are examined with the radiative transfer equation (RTE). From [65]:
∂I cos(θ) = κIb − βI (10–1) ∂z Z Z σ 2π π + s p(θ, φ, θ0, φ0) (10–2) 4π 0 0 × I (θ0, φ0, z) sin(θ0)dθ0dφ0 (10–3)
where I is intensity, β is extinction coefficient, κ is absorption coefficient, and σs is scattering coefficient. p is the scattering phase function. This can be simplified using axial symmetry. In addition, since at microwave frequencies, thermal emission is very low in comparison to the incident power, it is neglected in the following analyses.
Z ∂I $ +1 µ = −I + p(µ, µ0)I (µ0, τ)dµ0 (10–4) ∂τ 2 −1 µ = cos(θ) (10–5) Z z τ = β(z 0)dz 0 (10–6) 0
126 The boundary condition at z = 0 is
I (0, µ) = 0 (10–7)
for −1 ≤ µ ≤ 0, with the incident radiation
I (0, µ) = πF δ(µ − µ0) (10–8)
where µ0 is the direction cosine of the incident beam. There is a lower boundary, with reflectance. In terms of a boundary condition, this can be stated as:
Z 1 1 I (h, µ) = ρ00(µ, µ0)I (h, µ0)µ0dµ0 (10–9) 2 0 for −1 ≤ µ < 0.
10.3 Solution Details
The solution technique used here is finite difference (in τ) with Gaussian quadrature
(in µ). In Gaussian quadrature, intensity I is discretized at specific µj , the quadrature R points, and Idµ is approximated as a weighted sum over these muj , Σj aj Ij . Applying finite difference approximation,
I − I − µ i,k+1 i,k 1 + I = Σ a p(µ , µ )I (10–10) i 2∆τ i,k j j i j j,k
where i and j are angle indices and k is the optical depth discretization index.
For a purely absorbing medium, an isotropically scattering medium, and a Rayleigh scattering medium, respectively, the phase functions are
p(µi , µj ) = 0 (10–11)
p(µi , µj ) = 1 (10–12) P (µ )P (µ ) p(µ , µ ) = 1 + 2 i 2 j (10–13) i j 2
127 The upper boundary condition becomes
Ii,0 = 0 (10–14)
for µi < 0
ai Ii,0 = F (10–15) for µi = µ0. The lower boundary condition, for diffuse reflectance, becomes
R Σ a I = I (10–16) π j j j,K i,K
for µi < 0 and µj > 0. For specular reflectance, this becomes
RIj,K = Ii,K (10–17)
for µi = µ0 and µj = −µ0. 10.4 Physical Properties
Since microwave power transfer schemes described in 10.1 are less than 10GHz,
properties will be considered in this range, specifically at 2.45 GHz. The properties are considered homogeneous, and the atmospheric thickness is taken as 10 km. 10.4.1 Soil
The microwave reflectance of soil is a topic well-covered in the remote sensing
literature. Two famous papers are [66] and [67]. [66] discusses the effects of roughness,
soil moisture, and angle of incidence on reflectance and [67] formulates a widely-used model for the dielectric of soil. Their semiempirical model gives the soil dielectric as:
ρ α b α − β α − ²soil = 1 + (²s 1) + mv ²fw mv (10–18) ρs 2 ²s = (1.01 + 0.44ρs ) − 0.062 (10–19)
128 where α and β are fitting parameters, ρs and ρb are solid and bulk densities, mv is volume water fraction, ²fw is free water dielectric. At a frequency of 2.45 GHz, and 0.3 volume moisture fraction, the dielectric constant of sandy loam soil is about 18 + j3.
While there are multiple ways of varying degrees of complexity to characterize the soil’s reflectance, the two most straightforward are to treat it as specular or as diffuse. For specular reflectance, using the Fresnel formula, the reflectance is 0.6112.
When considering the SPS, the reflectance will be considered 0 because the downwelling beam is striking a rectenna array and not soil. 10.4.2 Atmosphere
The SPS involves radiation through the atmosphere, including the ionosphere.
Experiments [68, 69] show that microwave transmission at high frequency nonlinearly excites various electrostatic plasma waves; subsequent numerical simulation [70] of using particle-in-cell modeling shows that there is three wave coupling, where the transmitted wave serves as a pump wave, a backscattered wave occurs with a slightly lower frequency, and a much lower frequency electrostatic wave causes electron heating. Where the geomagnetic field is parallel to the transmission beam, the electrostatic wave leads to heating which prevents formation of the three-wave coupling and transmission efficiency is maintained at 90% or higher, while where the field is perpendicular the three-wave coupling continues periodically, causing most (80%) of the energy to be consumed in the coupling process. [71] shows that the ionosphere would have minimal effect on transmission through the atmosphere so the effects of atmospheric plasmas are not considered here.
For the microwave radiative transfer, there are three atmospheric components which participate noticably: gaseous water vapor, water droplets, and ice crystals. 10.4.2.1 Gaseous water vapor
[72] gives the dielectric of water vapor as a function of frequency and concentration as well as measurements. From this the absorbtion coefficient can be deduced. [73]
129 shows the impact of atmospheric water vapor (among other things) on microwave remote sensing of soil moisture at 19 GHz and above. [74] gives an RT model for microwave transfer in cirrus clouds. At high GHz frequencies, water vapor is strongly absorbing enough to be a significant loss at atmospheric concentration, but not at low GHz. From these studies, we can ignore the effects of water vapor for radiative WPT through the atmosphere. 10.4.2.2 Water droplets
Water droplets can play a significant role in microwave scattering and absorption. [63] perform a complex microwave radiative simulation of an evolving cloud using a different, time varying drop size distribution and a Henley-Greenstein scattering phase function. [75] examines microwave WPT using a two-stream polarimetric model with vertical and horizontal polarizations. [76] describes a radiative transfer model using different exponential drop size distributions for clouds or rain, combined with a frequency dependent dielectric function. They give scattering and absorbing coefficients at a range of frequencies.
The drop size distribution is a modified gamma (with units of µm−1/cm3):
P Q p(a) = K1a exp(−K2a ) (10–20)
−13 where for clouds the parameters are K1 = 1.26 × 10 , K2 = 0.75, P = 15, and
−28 Q = 1; for rain K1 = 7.41 × 10 , K2 = 0.025, P = 10, and Q = 1. They give the dielectric as
82.5 ² = 5.5 + (10–21) drop 1 + j0.0359/λ = 81.47 − j22.27 (10–22)
130 At low frequencies 2πa/λ << 1 so Rayleigh scattering dominates. The scattering and abssorbing cross sections for Rayleigh are ¯ ¯ 8¯m2 − 1¯ C = πa2 ¯ ¯x 4 (10–23) sca 3 m2 + 2 ³ ´ m2 − 1 C = −4πa2= x (10–24) abs m2 + 2 Z ∞
κ = p(a)Cabs (a)da (10–25) 0 Z ∞
σs = p(a)Csca(a)da (10–26) 0
−15 Numerically integrating over the particle distribution gives values of σs = 1.96×10 and κ = 1.10 × 10−8 for clouds and 1.39 × 10−11 and 6.88 × 10−9 for rain, with units in cm−1.
Figure 10-5. Cloud droplet distribution and absorption cross section.
Figs. 10-5-10-8 show the cross sections and distribution functions for clouds and rain.
131 Figure 10-6. Rain droplet distribution and absorption cross section.
10.4.2.3 Ice crystals
Atmospheric ice crystals are strong scatterers of microwaves. They occur in a wide range of shapes including spheres, plates, needles, and dendritic shapes, so their scattering properties vary widely. [77] examines the single-scattering, polarimetric effects of different ice crystals using polarimetric Rayleigh scattering and a gamma size distribution. [78] uses a Discrete Dipole Approximation for polarimetric, azimuthally dependent ice scattering for many shapes. Using the modified gamma with K1 =
−11 1.99 × 10 , K2 = 0.01, P = 2, and Q = 1, with the dielectric evaluated at 2.45 GHz [79]
97 ² = 3 + (10–27) ice 1 + j2πf × 57 × 10−6 = 3 − j1.12 × 10−4 (10–28)
−14 −13 gives σs = 7.45 × 10 and κ = 2.07 × 10 at 2.45 GHz.
132 Figure 10-7. Cloud droplet distribution and scattering cross section.
Figs. 10-9-10-10 show the cross sections and distribution functions for ice.
10.4.3 Vegetation
For the RF-harvesting sensor example considered in this chapter, the primary concerns are absorbtion and scattering by vegetation. This has been heavily investigated by the microwave remote sensing community.
Vegetation scatters and absorbs microwaves. Since there is no ”typical” vegetation, often this is calculated with empirical relationships. [80] computes single-scattering albedo and optical depth for corn and alfalfa at X (7-12.5 GHz) and Ka bands (20-30GHz) and shows a dependence of optical depth on plant water content. [81] develops a microwave dielecric model for vegetative tissues which plays an important role in subsequent studies of microwave-plant interactions. To relate this dielectric to absorption, sometimes the water cloud model is used [82]. [83] takes the semiempirical
133 Figure 10-8. Rain droplet distribution and scattering cross section.
approach further by incorporating another biophysical parameter, leaf area index. For this chapter, a 2m corn canopy will be considered, with τ = 1.8 and $ = 0.08.
Other papers take a more electromagnetically rigorous approach. At L-band, [64] treats the trunks in a forest stand as dihedral reflectors, and the canopy as volume of water droplets and incorporates multiple reflections. [84] incorporate scattering from leaves (disks) and stems (cylinders) oriented randomly in a fully polarimetric simulation. Measurements and modeling of scattering properties of vegetation with different leaf/stem geometries are given [85].
10.5 Results and Discussion
The RTE is solved for parameter values as determined from the literature review.
All simulations are done with 32 quadrature points and 8 discretization points in the τ direction. All intensities are shown in dBW/sr to highlight differences at low intensity levels.
134 Figure 10-9. Ice sphere distribution and absorption cross section.
Table 10-1. Parameter values for RTE. Parameter Cloud Rain Ice Vegetation
µ0 1.00 1.00 1.00 0.71 κ 1.10×10−8 6.88×10−9 2.07×10−13 9.00×10−3 −15 −11 −14 −4 σs 1.96×10 1.39×10 7.45×10 7.83×10 $ 1.78×10−7 2.01×10−3 2.60×10−1 8.00×10−2 −2 −3 −7 0 τh 1.10×10 6.90×10 2.81×10 1.80×10 R 0.00 0.00 0.00 0.61
10.5.1 Atmospheric Loss Estimation for Solar Power Satellite
From the literature review, the dominant methods of media particpation through
the atmosphere are Rayleigh scattering and absorbtion by clouds, rain, and ice. For all three scatterers, where µ < 0 (upwelling radiation) there is a drop in intensity due to the non-reflecting lower boundary; comparatively little radiation intensity is directed upwards, and that which is is due entirely to Rayleigh scattering. At the lower boundary,
135 Figure 10-10. Ice sphere distribution and scattering cross section.
the upwelling radiation (τ = τh and µ < 0) is zero, which is −∞ in log scale so it doesn’t show up on the log intensity plot.
Fig. 10-11 shows the intensity distribution in a cloudy atmosphere. The single-scattering albedo is very low (see Table 10-1) so the atmosphere is primarily absorbing.
Fig. 10-12 shows the intensity distribution in a rainy atmosphere. The single-scattering albedo is small but higher than for clouds as the mode particle size is larger so the atmosphere is more strongly scattering. This manifests itself in the intensity field as higher intensity levels off of the main beam compared to for clouds.
Fig. 10-13 shows the intensity distribution in an icy atmosphere. The single-scattering albedo is much higher than for clouds or rain. However, the magnitude of the σs and κ are quite small due to the fact that the mass density of particles is 1-2 orders of magnitude lower than for clouds and rain, and the dielectric for ice is much lower and less lossy than for liquid water. Hence the optical depth is much lower for the same
136 Figure 10-11. Log intensity distribution through cloud.
vertical distance. This manifests itself in the intensity field as higher intensity levels off of the main beam, due to scattering, but less loss, due to the dielectric of ice compared to water. 10.5.2 Loss Estimation for Radiofrequency-Harvesting Sensor Under Vegetation Canopy
Vegetation is handled as an absorbing and isotropically scattering media, with empirical parameters from the literature.
Fig. 10-14 shows the intensity field in the vegetative medium for a diffuse and a
specularly reflecting boundary. Since the optical depth is much higher for a vegetation canopy, the intensity drop is much higher. For the diffusely reflecting boundary, where µ < 0, the upwelling radiation is spread out. For the specularly reflecting boundary, where µ < 0, the upwelling radiation is a distinct beam. The single scattering albedo has the effect of increasing the off-beam intensity.
137 Figure 10-12. Log intensity distribution through rain.
10.5.3 Flux
Integrating over the solid angles at each z coordinate and normalizing to 1 W/m2 incident gives the flux at each height that would be received. The profiles are shown in Fig. 10-15. Using these the transmission efficiency can be estimated: 98.9% for clouds,
99.3% for rain, 99.999% for ice, 3.49% for vegetation with specular soil, and 8.96% for vegetation with diffuse soil.
10.6 Conclusions
This chapter examined wireless power transmission in the microwave regime. The far-field, radiative modes of wireless power transfer (WPT) through clouds, rain, ice, and vegetation were studied using a numerical solution of the equation of radiative transfer. The primary conclusion is that the optical depth due to atmospheric particles is much lower than that for vegetation canopy. Atmospheric ice has less attentuation than
138 Figure 10-13. Log intensity distribution through ice. atmospheric water droplets due to the different dielectric constants of frozen and liquid water. Higher single-scattering albedo increases off-beam intensities.
Ultimately, the SPS cannot be considered impractical due to transmission efficiency.
Other technical challenges such as the satellite construction and launch, and the efficiency of the receiving array, may limit the project. RF harvesting, relying on low power levels, is much more tolerant of the low transmission efficiencies associated with transfer through urban and rural environments. Far-field WPT has applicability, just as near-field and midrange power transfer.
139 Figure 10-14. Log intensity distribution through vegetation with diffuse and specular lower boundary.
Figure 10-15. Flux profile through different media.
140 CHAPTER 11 CONCLUSIONS
A proliferation of battery-operated devices, from cellphones to electric cars, has created a demand for a wireless charging system. The challenge of wireless power transfer (WPT) can be handled, broadly speakng, in three regimes. In near-field WPT, the transmission distance is much less than the characteristic dimension of the transmitter, and magnetic flux from one coil induces current in the receiver. Mid-range nonradiative WPT occurs when the transmission distance is one to several times the characteristic dimension, and power is transferred by means of slowly-decaying evanescent modes between two high-Q resonant structures. Far-field WPT is radiative in nature and is at transmission distances greater than the Rayleigh distance.
This dissertation has presented several aspects of the design of a WPT system.
The power electronics, electromagnetic, detection and estimation, and radiative transfer aspects of wireless power system were considered. Chapter 2 described the theory behind, and derived design equations for, a class E driving circuit and impedance transformation network. Chapter 3 derived relevant electromagnetic quantities for a near-field system. In Chapter 4, a coil design providing even fields was developed.
Chapter 5 extended the system to include multiple transmitting coils in parallel. Coil design for multiple transmitting coils in parallel is discussed in Chapter 6. Chapter
7 discussed the use and evaluation of ferrite shielding and found a suitable material candidate. Chapter 8 showed the development and testing of a Bayesian tracking algorithm for receiver discrimination and charge status determination. The extension of the system and coil design to midrange was presented in Chapter 9. Chapter 10 described the use of radiative transfer modeling for estimating losses of far-field wireless power transmission.
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148 BIOGRAPHICAL SKETCH Joaquin Casanova was born in Gainesville, Florida. Some stuff happened. Then, in 2006, he got his bachelor’s in agricultural and biological engineering from University of Florida, with a focus on agrisystems engineering and a senior project designing a thermally regulated table for seed germination studies. In 2007, he received the master’s degree in the same subject for his work on microwave remote sensing of soil and vegetation. After transferring to the UF Electrical and Computer Engineering Department, he earned another master’s degree in 2008, designing a three-dimensional fractal heatsink antenna. Shifting research focus to wireless power transfer, with a brief aside into high-voltage plasma generation electronics, he obtained his doctorate in 2010.
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