Transfer-Power Measurement Using a Non-Contact Method for Fair and Accurate Metering of Wireless Power Transfer in Electric Vehicles

Wireless power transfer (WPT) is emerging as the preeminent way to charge electric vehicles, but there appears to be no fair way to measure the power transfer. In this article, Faraday coil transfer-power measurement (FC-TPM) is presented. FC-TPM employs non-contact, open-circuited sense coils to measure the electromagnetic field from WPT and calculates the real power propagating through the air gap between the transmitter and receiver coils. What is measured is the real electromagnetic power, representing the pure dispensation of energy that unambiguously demarcates the losses on either side. FC-TPM was demonstrated to be 0.1% accurate in hardware over an Rx coil misalignment of up to 10 cm using a 1-kW WPT system. Fair metering incentivizes businesses and individuals to make choices that conserve energy and advance technology by providing more information and by properly assigning the financial loss. This article is accompanied by a video highlighting the essential contributions of this article.


I. INTRODUCTION
T RUST in technology is essential for adoption. Our society accepts what is fair, safe, and robust to use, where accurate measurement provides the discernment of those values. This article presents a fair and accurate measurement method for wireless power transfer (WPT).
Fairness in metering WPT has an integral significance to providers and consumers of energy as an arbiter in their competing financial interests. By 2030, EVs will consume over 1000 Manuscript  terawatt-hours of electricity every year worldwide; even a 1% misrepresentation in metering will cost energy consumers and providers over $ 1 billion 1 [18]- [21]. For fairness, the cost of lost energy must be appropriately assigned among the stakeholders. Transmitter losses must be disaggregated from receiver losses for the equitable metering of WPT. Energy station owners and EV owners will then be individually motivated to improve their efficiency and hence reduce their financial losses.
The proper demarcation line for the "point of sale" in wireless charging had been posited to be physically between the transmitter (Tx) and vehicle receiver (Rx) coil by [22]. 2 Transfer-power (P Transfer ) is the missing link. It is the real power through the air gap, purely dispensed from the Tx coil to the Rx coil. This real power can be represented electromagnetically through the Poynting vector illustrated in Fig. 1.
At first glance, there appeared to be no direct way to measure the quantity of power transfer; rather, what had been currently available were only the voltages and currents at the Tx and Rx electrical terminals with which to calculate the Tx input 1 Based on EV projections: 1) 250 million EVs by 2030 [18]; 2) 4500 kWh/year/EV [19]; 3) $0.1/kWh, the 10-years average price of electricity to ultimate customers in the transportation category [20]. 2  (P Tx ) and Rx output (P Rx ) power, which had been performed to varying degrees of accuracy and robustness in [23] and [24]. The shortcoming of these methods is that Tx input and Rx output power necessarily incorporate aggregated losses from both sides, including those from winding resistance and eddy currents [25], [26].
Except for the unlikely case that the losses are symmetric in Tx and Rx, the correct attribution of power inefficiency is not possible. Fig. 2 illustrates the conundrum: Input or output power measured at the Tx or Rx electrical terminal imposes the cost of the aggregated losses to one side or the other unilaterally; for example, if power is measured at the Tx terminal (P Tx ), metering adds Tx losses to the evaluation of transferred power, which ought to have been excluded in pricing, and thus, customers are overcharged. Likewise, metering at the customer (Rx) terminal (P Rx ) subtracts Rx losses from the evaluation of transferred power and hence represents an undercharge. These inequities are exacerbated in systems, where low efficiency from cost-cutting in design, production, installation, or maintenance is incentivized without proper metering, with another potential for abuse when there is physical access to the measurement terminals. Even if power efficiency is legislated, a robust way to validate disaggregated efficiencies is still needed.
These flaws can be overcome by measuring the transfer-power through the air gap between the Tx and Rx coils, as shown in Fig. 2. Not only will this provide fair metering, but will also disaggregate individual Tx and Rx efficiencies, P Transfer /P Tx and P Rx /P Transfer , respectively. As a diagnostic, it can financially incentivize the decisions and behaviors of providers and customers.
The subject of this article, Faraday coil transfer-power measurement (FC-TPM), originated in the following conference papers [27], [28], which to our knowledge is a first among methods in directly metering wireless charging of EVs. This journal article significantly expands on the original conference papers by the following: 1) providing a direct connection to the Poynting vector; 2) a thorough decomposition and analysis of the power balance that includes eddy current losses from cross-coupled magnetic fields; 3) FEM analyses for losses; 4) performance analysis of coplanar sense coils; and 5) verification on kilowatt-level WPT hardware.
FC-TPM employs non-contact open-circuited sense coils that are electromagnetically coupled to the Tx and Rx coils from which transfer-power is reconstructed. These sense coil voltages directly map to the Poynting vector, which is the directed power density, as shown in Fig. 1. FC-TPM is analogous to trusted third party gasoline fuel pump metering today from an arbitration perspective with inspection performed by an unbiased third party: for example, the Michigan Department of Agriculture's Weights and Measures Program as illustrated in Fig. 3.
The rest of article is organized as follows. Section II defines the transfer-power from the Poynting vector and presents a formulation based on a transformer model. Section III presents the transfer-power in lossy WPT coils from a winding loss model. Section IV presents FC-TPM: We derive the mapping of the sense coil voltages to the Poynting vector to validate FC-TPM as a fundamental measurement of WPT's real power flow and also show why employing multiple sense coil voltages makes FC-TPM accurate over misalignment between the Rx coil and Tx coil. Section V discusses how FC-TPM sense coils can be designed to be both physically and electromagnetically unobtrusive. Section VI demonstrates accurate FC-TPM in hardware over a standardized Rx coil misalignment of up to 10 cm with a 1 kW WPT system. Finally, Section VII is a summary of the main contributions of this article towards future research.

II. TRANSFER-POWER DEFINED FROM THE POYNTING VECTOR IN WIRELESS POWER TRANSFER
In WPT, we define transfer-power to be the real power propagated from a transmitter (Tx) coil to a receiver (Rx) coil through the intervening space. In this section, the formulation for transfer-power is derived through both the Poynting vector and a transformer model; additionally, we show how these two formulations are equivalent and directly map to each other.

A. Transfer-Power From the Poynting Vector
P Transfer or transfer-power can be defined from the Poynting vector, which is the directed electromagnetic power density and is determined by the cross product of the electric field E with the magnetic field H [29]. The average power p avg , which is the real transfer-power, can then be obtained by a surface integral of the real part of the time-averaged complex Poynting vector S where Note that, E and H are the phasors of time-harmonic fields whose magnitude is the rms value. Re{·} indicates the real part and the asterisk ( * ) indicates the complex conjugate. The Tx and Rx coils can be approximated as circular current loops whose basic physics can be illustrated by magnetic dipoles [30], as shown in Fig. 4 to derive the Poynting vector. The radii R x of the Tx and Rx current loops are identical and assumed to be much smaller than the distance between the two coils d T :R (R x d T :R ). The Poynting vector through the point Q : (r, θ, ϕ) in spherical coordinates is analyzed on the infinite plane P, which is located at z = d T :R /2 between the Tx and Rx coil, as shown in Fig. 4, where the magnetic field H and electric field E created by the magnetic dipoles were formulated in [30] and [31] in spherical coordinates. However, the Poynting vector and the resulting average power calculation can be represented more simply in cylindrical coordinates; the Poynting vector is derived from E and H where A = πR 2 x is the Tx and Rx loop area, and I T and I R are the corresponding currents. Note that in cylindrical coordinates (ρ, ϕ, z) The real part of the time-averaged complex Poynting vector S is then Note that only theẑ component of the Poynting vector contributes to the real power transfer, where both the Tx and Rx coil currents I T , I R determine the magnitude and direction of the Poynting vector. Fig. 5 shows the time-averaged Poynting vector field, simulated by the finite element method (FEM) in COMSOL; power is transferred from the Tx coil to the Rx coil when the Tx coil current leads the Rx coil current (i.e., the Tx and Rx coil current phase difference is positive, θ T − θ R > 0). Particularly, there is maximum power transfer when the phase difference is 90 • . The transfer-power P Transfer is the average power, which is calculated from (1), applying the surface integral to (5) over the infinite plane P P Transfer = p avg (6) P Transfer , the real power leaving the transmitter, is the real part of the product of the complex conjugate of the Tx coil current I * T and induced voltage jωM R:T I R (from the Rx coil current), where the geometric factor is the mutual inductance M R:T from the Rx to the Tx coil, which also appears in the transformer model.

B. Transfer-Power in the Transformer Model
The Poynting vector illustrates how WPT operates. In practice, directly working with the Poynting vector is cumbersome; a transformer model is both elucidating and useful for the analysis and design of WPT.
Transfer-power can be formulated through a transformer model where losses are treated extrinsically. When a Tx and Rx coil pair are magnetically coupled through an air core, as shown in Fig. 6 and when there are no winding and eddy current losses, the Tx and Rx coil voltages (V T and V R ) are where L T and L R are the self-inductances of each coil; M X:Y is the mutual inductance from coil X to coil Y ; ω is the angular frequency; and {V X , I X } are the phasors whose magnitude is the rms value. Note that M R:T = M T :R because of reciprocity. In this article, the subscripts attribute each variable to a particular coil: T (Transmitter), R (Receiver), natural numbers 1, 2, . . . (sense coils); colons (x : y) indicate a parameter relationship from coil x to coil y. In a WPT system with lossless coils, 3 the transfer power is equal to the real power at the coil terminals. The real power at the terminals of the Tx coil in this case is  Note that the combination of the incident magnetic field from the Tx coil current I T and the reflected electric field from the induced voltage jωM R:T I R from the Rx coil current I R comprises transfer-power [30], [32]. The geometric factor in (7) is the mutual inductance M R:T . The transfer-power can also be similarly derived from the Rx side as Re{jωM T :R I T I * R }, which results in a negative value of P Transfer , indicating that power is consumed by Rx.

III. TRANSFER-POWER IN LOSSY WIRELESS POWER TRANSFER COILS: WINDING LOSS BREAKDOWN
The principal transfer-power, which we denote P Tx:Rx , corresponds to the directed real power from the terminal currents described in Section II. However, for metering, principal transferpower may not offer a complete description of transfer-power when cross-coupled loss mechanisms are significant.
Winding losses decompose into ohmic and eddy current losses, which we derive in this section. These losses manifest as heat dissipation in either the Tx or Rx coil, which is formulated from the coil (Tx or Rx) currents. Assignment of losses based on heat dissipation is consistent with the Poynting vector definition of transfer-power, which is the power flow through the intervening space between the Tx and Rx coils. Transfer-power is saliently different from the black box notion of input and output electrical terminal power.
Fair metering using transfer-power means that costs for power losses are imposed on the transmitter (service station owner) and the receiver (EVs owner) equitably based on the amount of the each side's physical power dissipation, which manifests as heating. This heat dissipation-based demarcation of the loss penalizes stakeholders who use inferior quality coils, power electronics, or other system components that cause loss. In other words, fair metering motivates providers and customers to advance their system (e.g., by using better litz wire or winding methods) to reduce losses.

1) Input and Output Terminal Power:
The input power P Tx and output power P Rx that are measured at the coils' electrical terminals are where V T , V R : Respective Tx, Rx coil terminal voltage I T , I R : Respective Tx, Rx coil current.
The terminal voltage contains not only the induced coil voltages, represented by (8) and (9) but also the voltage drops that are related to winding losses, as shown in (89) and (90) in Appendix A. From the perspective of power conservation, 4 the difference between the Tx coil input power and the Rx coil output power is the aggregate power loss within both the Tx and Rx coils.
2) Winding Losses: The main purpose of our winding loss derivation and ensuing loss breakdown is to identify the source of each loss and to clarify where each loss is dissipated. This is especially important for WPT metering in that the Tx and Rx coils are magnetically coupled, where each coil's current and hence magnetic field can generate a loss in the other's coil.
Using a winding loss model 5 where eddy currents can be represented by an additional winding on a transformer, we can further show that the measurement of transfer-power disaggregates the losses between the Tx and Rx coils properly in contrast to the input and output terminal power, which lump both losses.
We decompose the winding losses into (i) ohmic losses due to ac and dc winding resistance and (ii) eddy current loss, which is the loss from eddy currents within a coil due to the external proximity effect, where the opposing coil generates an external magnetic field: For example, the external proximity effect loss in the Rx coil due to the Tx coil current and resulting magnetic field. We refer to this external proximity effect loss as eddy current loss in this article.
Winding losses can be represented by the source currents P W,Tx represents the Tx winding loss, measured at the Tx coil terminal as a part of the input power; R T is the effective winding resistance, which is frequency-dependent and accounts for the skin and internal proximity effects when there is no external field. The discussion of γ R is identical to that of γ T because the transmitter and receiver can be interchanged [for example, (15) and (16) are symmetric in the sense that R can replace T in the subscripts or vice versa to obtain the other equation, hence also reflecting the symmetry of the physics]. γ R encapsulates the effect of geometry in how the external field from the Rx coil affects the current density distribution in the Tx windings. The external field can reinforce or weaken the self-field depending on its magnitude and relative phase, which is represented in I * T I R . The external field can also reinforce or weaken the self-field depending on the relative directions and relative strengths of the fields, which is represented in γ R and therefore a factor which depends on geometry. γ T and γ R can be different because the Tx winding and the Rx winding are not necessarily identical; therefore, the geometric effect of the opposing external magnetic field is not necessarily symmetric. The Tx coil current also contributes to the loss because of the external proximity effect, which is dissipated in the Rx coil; 6 eddy currents are created in the Rx coil from the time-varying magnetic field from the Tx coil current. The eddy current can be modeled as an independent winding [33], [34] that is electromagnetically coupled to the Tx coil. The eddy current loss is then a real power transferred from the Tx coil to the eddy winding and can be represented by the source current (Tx coil current) and the effective resistance R T :r . Appendix A introduces the eddy winding model and derives the eddy current losses dissipated in the Rx coil. R T :r is the effective resistance that encapsulates: 1) the magnetic coupling between the Tx coil and an eddy current winding in the Rx coil, and 2) the effect of the impedance in the eddy current winding. P W, Rx represents the Rx winding loss, measured at the Rx coil's electrical terminal as a part of the output power, where R R , γ T , and R R:t are defined in the same manner as for the Tx winding loss. Note that γ R R T and γ T R R are equal because of reciprocity.
Input power P Tx in (13) and output power P Rx in (14) can be formulated from the terminal voltages (89) and (90) of the winding loss model in Appendix A  (12); 2) P Loss, ohmic (Tx) , ohmic losses due to the Tx coil winding resistance R T ; and 3) P Loss, eddy (Rx) , the external proximity effect loss, dissipated in the Rx coil, generated by the Tx coil current I T . At the Rx coil, P Tx:Rx is received and losses are incurred, which are composed of: 1) P Loss, ohmic (Rx) , ohmic losses due to the Rx coil winding resistance R R ; and 2) P Loss, eddy (Tx) , the external proximity effect loss, dissipated in the Tx coil, generated by the Rx coil current I R . The output power P Rx measured at the Rx coil terminal is then the difference between P Tx:Rx and the incurred loss. Fig. 7 shows the power flow in WPT. Observe that input and output terminal power commingle winding losses and therefore cannot disaggregate losses properly: 1) Input power P Tx aggregates winding losses, which are the ohmic losses dissipated in the Tx coil P Loss,ohmic(Tx) and the eddy current losses dissipated in the Rx coil P Loss,eddy(Rx) and 2) output power P Rx also aggregates winding losses, which are the ohmic losses dissipated in the Rx coil P Loss,ohmic(Rx) and the eddy current losses dissipated in the Tx coil P Loss,eddy(Tx) . For example, when the Rx coil is open-circuited, the output power P Rx is zero, but there is heat dissipation in the Rx coil from the Tx coil current, which should be attributed to the Rx coil. It is worth noting that the transfer-power that is useful for metering comprises the principal transfer-power and the eddy current losses P Transfer = P Tx:Rx + P Loss,eddy(Rx) − P Loss,eddy(Tx) .
For the purpose of fair metering, we can redistribute power losses based on the Tx and Rx coils' dissipation P d,Tx and P d,Rx (21) which are in contrast to (15) and (16). Note that what is dissipated in the coil is a combination of 1) the ohmic loss and 2) the external proximity effect (eddy current losses). Equations (20) and (21) explain the eddy current losses in the Tx and Rx coils. This manifests in the example where despite the Rx coil being open-circuited (I R = 0), P d,Rx is nonzero, yet with a loss generated by the Tx coil current.
Ohmic loss from the Tx winding resistance R T Eddy current losses in the Rx coil, generated by I T Eddy current losses in the Tx coil, generated by I R Output power at the Rx coil .
Note that the negative sign before P Rx accounts for the power consumption in the Rx coil. Fig. 8 illustrates the conservation of power in WPT.

B. Transfer-Power Metering With Losses
In Section II-A, transfer-power is defined from the Poynting vector, which is the electromagnetic power density through space. As shown in Fig. 7, the transfer-power P Transfer for lossy coils is the sum of three power flows: 1) P Tx: Rx : the real power transferred from the Tx to Rx coils, Re{jωM T :R I R I * T } in (12), which is the principal transfer-power; 2) P Loss,eddy(Rx) : eddy current loss in the Rx coil; and 3) P Loss,eddy(Tx) : eddy current loss in the Tx coil, so that Metering based on transfer-power explicitly attributes a particular coil's losses as the power it is dissipating as heat, which is consistent with the assertion of the Poynting vector definition. Transfer-power is equal to transmitter coil terminal power minus the Tx coil power dissipation In other words, transmitters Tx (providers) automatically pay for the cost of P d,Tx if metering is based on transfer-power. Similarly, receivers Rx (customers) automatically pay for their own power dissipation P d,Rx along with the power they receive for consumption. From the receiver side, the transfer-power can be represented by For example, a customer can reduce their dissipation (from ohmic and transmitter-induced eddy current loss), hence their cost of energy, by using a higher quality litz wire.

C. Extraction of Winding Loss Model From FEM Simulations
We can calculate winding losses from FEM simulations; we can show how the power and loss distribute among the various mechanisms for various Tx and Rx coil configurations to study how, in practice, power partitions in WPT charging. Circular versions of wireless charging coils based on SAE J2954 [35] were evaluated using finite element analysis using COMSOL. Different transmitter and receiver coil radii (r T and r R ), air gaps (d T :R ), windings (N T and N R ), and power classes, representing a wide range of standardized options, were selected. Appendix B presents the details of the simulations and calculations, and the coil specifications for Table VI. Two-dimensional axisymmetric simulations were performed with the Tx and Rx coils driven by current sources. Although in practice, litz wire is typical, solid wire was used in these analyses as worst-case examples. The Tx coil currents were chosen to be the maximum current for each power class, and the Rx coil currents were selected to satisfy the maximum power level for the class, both specified in [35]. Table I presents the principal transfer-power P Tx:Rx , ohmic losses (P ohmic,Tx and P ohmic, Rx ), and eddy current losses (P eddy, Tx and P eddy, Rx ) as a percentage of the input power for each coil specification. As expected (24) and (25) give identical results for each coil configuration. The results are compared graphically in Fig. 9. The power flow from the Tx to Rx coil P Tx:Rx is the dominating quantity, while ohmic losses are the greatest portion of the winding losses. The worst case for the percentage eddy current loss was 1.12 %, dissipated in the Rx coil for the WPT2/Z1 class.

IV. FARADAY COIL TRANSFER-POWER MEASUREMENT
FC-TPM is a non-contact electromagnetic method to measure transfer-power through an intervening space by making , P ohmic,Tx , P ohmic,Rx , P eddy,Tx , P eddy,Rx are percentages of the input power P Tx . η is the Tx-to-Rx coil efficiency in percent, P Rx /P Tx × 100.  FC-TPM is the first among methods in EV charging to measure the power flow through space, resulting in fair metering. Advantages of FC-TPM include the following.
1) Accuracy that is independent of self-inductance and ohmic loss of Tx and Rx coils, power electronics (e.g., compensation circuit topologies), and electrical loads of the Rx side. 2) Accuracy that is insensitive to coil misalignment, operating frequencies, and various coil wire types (e.g., solid and litz wires). 3) Small footprint with sense coils that are few in number and diminutive in size whose electromagnetic and physical disturbance is negligible. This section shows how the sense coil voltages reconstruct the real power flow in WPT. First, we derive the Poynting vectors from the sense coil voltages to prove that FC-TPM directly measures the electromagnetic power flow through the air gap. Then, we derive the transfer-power from the sense coil voltages using the transformer-model, which reveals the benefit of using geometric parameters and leads to calibration strategies for FC-TPM.

A. Mapping the Sense Coil Voltages to the Poynting Vector
The Poynting vector can be represented by sense coil voltages from which transfer-power can ultimately be derived. Sense coils sample the electromagnetic field from which the Poynting vector can be reconstructed. If the sense coils are placed coaxially with the Tx and Rx coils, as shown in Fig. 11, a sense coil voltage V i is induced by theẑ component of the magnetic field from (4) Note that d X:i is the distance between coil X and sense coil i; R i is the radius of sense coil i; A = πR 2 x is the Tx and Rx loop area, where R x is assumed to be much smaller than the distance between the two coils d T :R (R x d T :R ) from the magnetic dipole approximation of the Tx and Rx loops. What results from (28) and (29) is that the Tx and Rx coil currents (I T and I R ) can be reconstructed by a linear combination of the pair of sense coil voltages (V i and V j ) As shown in (5), the Poynting vector can be represented by the Tx and Rx coil currents. In other words, determining the Tx and Rx coil currents from the sense coil voltages in (30) and (31) enables one to find the Poynting vector.
The real part of the time-averaged complex Poynting vector is where λ is the geometric parameter, determined by the sense coil positions. In summary, the Poynting vector at any point in the plane of interest can be mapped by the imaginary part of the complex conjugate pair of sense coil voltages Im{V i V * j }.

B. Theory of FC-TPM Using the Transformer Model
The sense coil voltages can represent transfer-power through the transformer model. In this and the following two sections (IV-C and IV-D), we neglect external eddy current losses. 7 Because there are no eddy current windings, the open-circuited sense coil voltages are induced only by the Tx and Rx coil currents. 7 This assumption elucidates the principle of accurate FC-TPM over the Rx coil's misalignment and corresponding calibration strategy in Section IV-C. The eddy current loss and hence the winding can be taken into account by increasing the matrix's dimension in (38) with four sense coils to include eddy currents I t , I r and corresponding mutual inductances, as shown in (63) in Section IV-E.
The sense coil voltages V i in the frequency domain are The Tx and Rx coil currents (I T and I R ) can be derived from two sense coil voltages (V i and V j ) where M T :i and M R:i are the mutual inductances from the Tx and Rx coils to the i th sense coil. The transfer-power in (12) is represented in terms of the mutual reactance ωM R:T between the Tx and Rx coils, and the Tx and Rx coil currents (I T and I R ). Sense coil voltages (V i and V j ) can, thus, represent the transfer-power where We denote k m:n as the coupling coefficient between any two coils m and n, where k m:n = M m:n / √ L m L n . κ ij is a function of the coupling coefficients between coils and the selfinductances of the sense coils (L i and L j ) at angular frequency ω. In the frequency domain, the calculated component of transfer-power at each frequency point can be converted to the average power in watts (p Transfer ) through Parseval's theorem [36] where N is the number of data samples and P Transfer is calculated from the discrete-time Fourier transform (DTFT) of the voltage and current signals. Total power in the time-domain is equal to that in the frequency domain.

C. Accurate FC-TPM Throughout Rx Coil Misalignment
Misalignment between the Tx (energy charging stations) and Rx coils (EVs) is unavoidable; even an autonomous system can have misalignment. In fact, SAE J2954 [35] certifies misalignment up to 10 cm for a 45 cm diameter coil. FC-TPM must therefore be accurate over that misalignment range, as illustrated in Fig. 12.
FC-TPM as metering is practical for energy service stations in that only sense coil voltages are needed for measurement during charging. In the previous Section (IV-B), the corresponding geometric parameters were required to be constant so that they could be calibrated in advance. As detailed in (41) and (42), the geometric parameter κ ij is needed together with the sense coil voltages to determine transfer-power. However, coil misalignment makes κ ij vary. This is so because κ ij is a function of coupling coefficients (k R:j , k R:i , and k R:T ) that vary over misalignment. 8 To resolve the problem of parameter variation, we employ multiple sense coils to collect more information, where nonvarying geometric parameters can be obtained over misalignment to determine the transfer-power. We found that a linear combination of pairwise-products of sense coil voltages determine the transfer-power accurately. The coefficients for the linear combination are geometric parameters that do not vary over misalignment. In fact, neither knowledge nor an explicit measurement of misalignment is needed to determine the transfer-power.
Despite the significant misalignment allowed for SAE J2954, the coupling coefficients from the Tx and sense coils to the Rx coil are well-approximated by quadratic functions (i.e., second-order polynomials) over the Rx coil misalignment. This quadratic approximation explains how a linear combination of pairwise-product of sense coil voltages can accurately determine transfer-power at any misalignment, with constant coefficients for the linear combination, hence allowing calibration. In fact, calibration requires neither knowledge nor explicit measurement of misalignment. 8 A practical design requires Tx and sense coils to be stationary making k T :i , k T :j constant over Rx coil misalignment. Tx-to-Rx; and 2) Rx-to-sense-coils. The geometric parameter κ 12 (x) (greendots) in (45) for two coplanar sense coils (sense coil 1 and 2, specified in Table II) are plotted over Rx coil misalignment. Red lines are the corresponding secondorder polynomial fits.
In this section, we show how the coupling coefficients can be approximated by quadratic functions of misalignment. We then present a formulation to determine transfer-power that is accurate over misalignment using multiple sense coil voltages.
1) Quadratic Approximation of Coupling Coefficient Variation Over Misalignment: When the coupling coefficient from the Tx to Rx coil varies quadratically over the Rx coil's misalignment, it is advantageous to choose sense coil positions and radii so that the coupling coefficients from the sense coils to the Rx coil are also quadratic dominant. It is important to note that the coupling coefficient functions are positive definite; the quotient of positive definite quadratic functions is also quadratic dominant, albeit over a narrower interval. Sense coils that are coaxially positioned with the Tx coil are especially good candidates for quadratic dominant coupling coefficients. In the following Section IV-C2, we show how least-squares optimization of a parameterization of coupling coefficients based on quotients that form a quadratic dominant function is particularly good at determining transfer-power.
We derive a quadratic approximation for the mutual inductance and hence the coupling coefficient from Grover [37] for two circular filaments with lateral misalignment. The quadratic approximation is a Taylor expansion with respect to lateral misalignment. In bounding the Lagrange remainder, we show that this second-order Taylor approximation is accurate over the misalignment range of interest, which we detail in Appendix C. In Fig. 13, we show the calculated results for the 1) the mutual inductance, and hence the coupling coefficient, and 2) the geometric parameter κ ij for the Tx and Rx coils using the numerical model for circular filaments in [38].
The mutual inductances and κ ij are very nearly quadratic with a coefficient of determination R 2 > 0.999. 9 We further investigated the quadratic dependence for different sense coil positions and radii using R 2 . Fig. 14 shows that sense coil positions closer to the Tx coil are better.
2) FC-TPM Formulation Over Rx Coil Misalignment: Sense coil voltages implicitly contain information about the Rx coil misalignment. When combined with functions that implicitly contain information about the sense coil positions and sizes, transfer-power can be determined with minimal error from misalignment.
The transfer-power represented by (41) and (42) is a function of the lateral misalignment x of the Rx coil where Note that V i,j (x) are sense coil voltages at a particular Rx coil misalignment x. We define V i , V j as the pairwise-product of sense coil voltages, where V i and V j are complex scalars. For power metering, the Tx and sense coils have fixed positions, whereas the EVs (Rx coils) drive in to charge; therefore, the coupling coefficients between Tx and sense coils (k T :i and k T :j ) do not change.
If we use (44) to determine transfer power, the misalignment x and every coupling coefficient as a function of x must be explicitly and accurately known. However, in using multiple sense coils, we can transform the overdetermined set of sense coil voltages to a function that determines transfer-power from these sense coil voltages alone. We will show that this function can be simply calibrated over the span of misalignment, but without needing a measurement of misalignment at all.
In using multiple sense coils, the pairwise-product of voltages from each unique pair V i , V j can be linearly combined and scaled by corresponding coefficients α ij i,j∈Q (47) where N is the number of sense coils.
As discussed earlier, the function κ ij (x) is well approximated by a quadratic polynomial If we choose α ij so that where i,j∈Q and perform a least-squares optimization to obtain α ij then the transfer-power can be determined despite misalignment from the linear combination of unique pairwise-products of sense coil voltages where are only voltage measurements and implicit functions of x; N is the number of sense coils. Note that the geometric parameters α ij , which are calibrated initially, are constant (independent of x).

D. Calibration of Constant Geometric Parameters Over Misalignment and FC-TPM Numerical Results
In this section, we explain how the geometric constants in Section IV-C2 can be calibrated. We then numerically evaluate FC-TPM over misalignment using well-known models from the literature.

1) Formulating the Calibration Matrix and Vector:
The calibration of the geometric parameters α ij requires sense coil voltage measurements and transfer-power data from a reference standard. The sense coil voltage measurements are combined as uniquely paired products in a data matrix . . .
The transfer-power corresponding to each row of W is contained in the elements of column vector p Each row of W corresponds to a particular measurement condition (e.g., misalignment, power, load, etc.). The number of unique pairings for the the pairwise-product of sense coil voltages from N sense coils is N w = N C 2 . For l data points of variation that consists of m data points of misalignment (x 1,...,m ) and n data points of Rx coil loading (Z 1,..,n ): l = mn, W ∈ R l×N w , and p ∈ R l×1 . The data for calibration needs to span the space of variation, which includes geometric variation (e.g., misalignment) and load. The variation in load needs to span the real impedances corresponding to the required measurement range for transferpower.
It is worth noting that the variation encapsulated in W and p does not have to be uniform, nor does the explicit domain of variation (e.g., actual misalignment displacement x) need to be measured.
The vector of geometric parameters α ij ∈ R N w ×1 for N sense coils is From (52) which is overdetermined, allowing the calculation, and hence calibration of α using the least-squares method 2) Coil Configurations: Two different placements of sense coils were considered for the numerical analysis, as shown in Fig. 12(a) and (b).
(a) Vertically stacked sense coils: The N sense coils are vertically stacked above the Tx coil at 1 cm intervals. The radii of the Tx, Rx, and sense coils are identically 25 cm. (b) Coplanar sense coils (low-profile): All N sense coils are concentric and placed on the same plane, which is 1 cm above the Tx coil. The sense coil radii varied from 26 to 23.5 cm, decreasing at 0.5 cm intervals. 3) Numerical Results: Mutual inductances were obtained from a well-known circular filament model [38] over the Rx coil misalignment x. The model assumptions include: i) Concentrated windings as shown in Fig. 12; ii) fundamental frequency only; and iii) no measurement noise. The Tx and Rx coils are driven by current sources. The transfer-power and sense coil voltages were calculated with (12) and (37), respectively, at each misalignment. The equivalent circuit for the numerical model is shown in Fig. 10(b). The geometric parameters α ij were calibrated using sense coil voltages and transfer-power; the accuracy of FC-TPM was then evaluated using leave-oneout cross-validation (LOOCV) [40]. 10 FC-TPM accuracy was evaluated for different numbers of sense coils (from two to six). For these numerical results, m = 11 misalignment data points (0-10 cm at 1-cm intervals) and n = 6 load data points were used to calibrate α ij .
The FC-TPM errors over misalignment were calculated for each data point using LOOCV. The percentage errors are calculated between the standardized value P Transfer (x m , Z n ), and the reconstructed valueP Transfer (x m , Z n )  Fig. 15(a) and (b) show the FC-TPM errors over misalignment for two different sense coil placements, as defined in Fig. 12. We plotted the worst-case absolute error percentages of FC-TPM where u k is the vector of parameter variations over which the error is calculated. The worst-case absolute errors at each lateral  Fig. 12(a). (b) Coplanar sense coils were placed coaxially, 1 cm above the Tx coil, as shown in Fig. 12(b). The radii of the sense coils decrease from 26 at 0.5-cm intervals. misalignment x were calculated and plotted in Fig. 15, where For both sense coil configurations, the errors were nearly constant over the misalignment with an increasing number of sense coils resulting in lower error. The lowest percentage errors were approximately 10 −6 % for six sense coils (green circles). Coplanar sense coils have a lower profile and hence are more practical for deployment in charging stations; these results show that coplanar sense coils have comparable performance to coaxial sense coils that are not coplanar. Sensor placement and sizing are analyzed over tradeoffs in Section V-B to corroborate the performance of coplanar configurations relative to other configurations.
Similarly, FC-TPM can be accurate over changes in the distance d T :R between the Tx and Rx coils. FC-TPM over the variations Δd T :R of the distance d T :R was numerically evaluated; Δd T :R ranged from −5 to 5 cm, resulting in the change in distance between the Tx and Rx coils from 15 to 25 cm, where the nominal distance d T :R was 20 cm. Coplanar sense coils were employed in Table II. FC-TPM accuracy was evaluated for 66 data points, which consisted of 11 data points of distance variations Δd T :R (-5 to 5 cm at 1-cm intervals) and 6 load data points. We plotted the worst-case absolute error percentage of FC-TPM at each Δd T :R in Fig. 16; the errors are nearly consistent over Δd T :R with an increasing number of sense coils resulting in a lower error. The lowest percentage errors are below 10 −4 % for six sense coils (green-circles).

E. FC-TPM With Eddy Currents, Rx Coil Misalignment, Different Litz-Wire Types, and Different Operating Frequencies
Eddy currents in the Tx and Rx coils not only add to losses, as presented in Section III but also change the magnetic field geometry by changing the current distribution in the coils nonuniformly. The eddy currents can create a magnetic field with nonnegligible coupling to the opposing coil. In other words, eddy currents induced on the Tx coil create fields that couple to the Rx coil and vice versa.
The induced eddy currents can be represented by additional windings in the transformer and the magnetic field geometry and hence coupling to different coils by the corresponding coupling coefficients. The variation in magnetic field geometry can be caused by a combination of 1) coil misalignment; 2) different wire types (e.g., litz-wire types); and 3) a range of operating frequencies.
In this section, we extend the formulation in Section IV-B and IV-C to include eddy currents and to show how a greater number of sense coils can maintain the accuracy of FC-TPM when there are variations in Rx coil misalignment. Through electromagnetic finite element simulation, we also show that a greater number of sense coils maintain FC-TPM accuracy despite the effects of different types of wire and operating frequencies.
1) FC-TPM With Eddy Currents and Misalignment: When there are eddy currents in the Tx and Rx coils, the sense coil voltages V i in (37) become V i = jωM T :i I T + jωM R:i I R + jωM t:i I t + jωM r:i I r .
(62) Note that I t , I r are the eddy winding currents in the Tx and Rx coils, as discussed earlier in Section III. M t:i , M r:i are the mutual inductances from the eddy windings to the ith sense coil.
For example, Tx, Rx, and eddy winding currents can be derived from four sense coil voltages V 1,..,4 ⎛ ⎜ ⎜ ⎜ ⎝ Detailed expressions for λ ij using an inverse matrix of the mutual inductances in (63) is presented in Appendix D. Note that a minimum of four sense coils N = 4 are needed to determine four coil currents (I T , I R , I t , I r ) and hence the transfer-power. If there are more than four sense coils, we can make use of the additional information by choosing four sense coil voltages at a time from the total of N sense coils, from which a total of N C 4 different formulations of (65) are constructed to determine the transfer-power. Each formulation is a linear combination of the pairwise-products of two sense coil voltages chosen out of the four sense coils in the formulation. Using (65), all the formulations can be used to determine the transfer-power, which is detailed in Appendix D. One can observe in Fig. 17 the effect of differing numbers of sense coils on FC-TPM when there are eddy current losses in the WPT coils. A 2-D axisymmetric FEM simulation in COMSOL was performed for two solid-wire WPT coils with the same dimensions as the hardware in Section VI. From (63) and (64), when eddy currents are significant, four sense coils are needed. Fig. 17 shows that with two-sense coils the errors are a considerable 0.66%. As the number of sense coils increases to four, the error reduces to a much smaller 1.6 × 10 −9 %. Additional sense coils beyond four do not significantly improve the error for this case where the WPT coils are aligned.
In particular, this collection of four sense coils from the N > 4 sense coils results in N C 4 independent transfer-power formulations, where we can extend the principle of FC-TPM over misalignment in Section IV-C to that which includes eddy current. By increasing the number of sense coils, the formulation in (65) can embed (52), ultimately resulting in a single set of geometric parameters that also include Rx coil misalignment together with the effects of eddy current.
The transfer-power at each misalignment x can be determined by where λ ij (x) can be approximated by an n th -order polynomial The linear combination of the N w transfer-power formulations with constant coefficients δ k is then which approximates to a form like (52) 11 Note that the geometric parameters λ (k) ij of the same pair of sense coils (i, j) for each subset s k are distinct (e.g., λ when δ k is optimized by choosing the appropriate sizes and positions of the sense coils so that which has (N −2) C 2 elements; in other words, G ij is a set of indices of s k , which includes a specific sense coil pair i and j, noting that s k is a set of combinations of four sense coils. 12 The coefficients α ij can be obtained from a least-squares optimization like (51) The transfer-power can then be determined over the Rx coil misalignment by a single set of geometric parameters α ij .

2) FEM Simulations of FC-TPM With Eddy Currents Over Multidimensional Variations: Misalignment, Litz-Wire Types, and Operating Frequencies:
The principle that enables accurate FC-TPM over variations (e.g., misalignment) is the use of an overdetermined set of sense coils whose geometric parameters are such that the linear combination of the pairwise-products of their voltages are insensitive to errors from the variations. These had been presented in Sections IV-C2 and IV-E1.
The optimal linear coefficients are calibrated through the least-squares minimization of a calibration set, which is appropriately chosen over the range of variations of interest, as shown in (53)- (55). This formulation can be expanded to a multidimensional simultaneity of variations, such as having EVs (Rx coils) with different types of litz wires and operating frequencies.
Calibration can be undertaken for both Tx and service station sense coils during manufacturing or when commissioned in the field for retrofits or repairs. 13 Three-dimensional FEM simulations in COMSOL were performed to demonstrate accurate FC-TPM over multidimensional variations in the Rx coil, where the effects of eddy currents are included in the FEM simulation. The goal of the simulations is to confirm that transfer-power can be determined accurately using only a single set of geometric parameters regardless of those variations. 13 From an arbitration perspective, with the inspection performed by an unbiased third party, the official trucks can have their sense coils, attached to the Rx coils to inspect the charging stations' metering accuracy. This is possible because the official trucks and sense coils are calibrated over Tx coil variations in standard laboratories. This is analogous to the Weights and Measures Program today in that officials bring trucks with provers that are carefully calibrated in standard laboratories [41], [42] to check the accuracy of gas dispensers. Notably, we show that using only a small number of wire types (including litz wire) for calibration, FC-TPM is accurate for a broad range of wire types that are not in the calibration set.
As illustrated in Fig. 18, FEM results consist of a total 105 data points, which are a combination of 1) Rx coil lateral misalignment: {0, 2.5, 5, 7.5, 10 cm}; 2) operating frequencies: {80, 85, 90 kHz}, which are within the mandated range of SAE J2954 [35]; and 3) seven complex permeabilities of different Rx windings: μ = 1 (solid-wire) and μ = {0.8, 0.6}, μ = {0, 0.2, 0.4}, where μ = μ − jμ , which can model different bundles and strands of litz wire, using an equivalent complex permeability model [43], [44]; it is worth noting that the range of μ and μ in Fig. 18 covers a broad range of wire types [43]- [45]. Fig. 19 shows an example configuration for a particular simulation iteration. To alleviate the computational complexity of an already intensive 3D simulation: 1) The Tx coil was modeled as a uniform surface current on a circular plane; 2) the Rx coil was a single-turn circular coil; 3) sense coil voltages were obtained postsimulation from the magnetic flux intersecting the coil area. The Rx coil radius was 12.5 cm, with a 5-mm diameter wire, specified in [35]. The Tx surface current's inner and outer radii were 5.5 and 12.5 cm, respectively. The air gap between the Rx coil and the Tx surface current was 10 cm. The sense coils were located on a plane 1.25 cm above the Tx surface current. The sense coil radii ranged from 18.25 to 3.25 cm decreasing at 1.5-cm intervals.
The accuracy of FC-TPM was verified using LOOCV. We used entire data subsets for a particular litz-wire type for validation. Specifically, in this particular variant of LOOCV, we iteratively left a particular litzwire type out of each calibration set and reserved it for validation to show that the calibration set can span the parameter variation space, hence maintaining FC-TPM accuracy over the broad range of litz-wire variations. In particular, the validation set consisted of 15 data points of a single wire type (i.e., a specific complex permeability), which were obtained from five misalignment values {0, 2.5, 5, 7.5, 10 cm}, and three frequency variations {80, 85, 90 kHz}. The remaining 90 data points consisting of 6 wire types, each with 15 variations, were the calibration set. The geometric parameters α ij were calculated at each LOOCV iteration for the calibration set using (53)- (57).
FC-TPM errors were calculated for each validation point. Using α ij with 11 sense coil voltages, the transfer-power was determined for each point using (70) and the error was calculated with (58), where the reference standard transfer-power was calculated from (25). Fig. 20(a) shows the worst-case absolute error percentage, as defined in (61), for each type of wire (represented by the complex permeability) as the number of sense coils is increased, where u k = [μ x k f k ]. Eleven sense coils reduced the error to below 0.1%. Fig. 20(b) shows the spread in error using 11 sense coils for each type of wire over frequency and misalignment variation; the errors were ranged from −0.083% to +0.054%.
Future work will include validating other variations including environmental influences for the one-time calibration of FC-TPM: 1) Variations in coil winding methods (e.g., concentrated, solenoidal, and spiral); 2) Tx coil variations in wire type and winding method; and 3) optimization of sense coil placement and size. Note that unacceptable environmental influences including hazardous objects or defects can be detected and power transfer stopped; for example, a foreign object, which is a fire hazard, can be detected by the FC-TPM system [46].

V. ELECTROMAGNETICALLY THIN AND PHYSICALLY FLAT SENSE COILS
Effective sensors should not affect that which it is measuring nor should it negatively impact the primary mission of charging vehicles. In this section, we show how the sense coils for FC-TPM can be constructed so they are a minimal electromagnetic perturbation and be designed to be below the pavement. Using high fidelity FEM using COMSOL, we show that the eddy current losses dissipated in the sense coils are insignificant and hence electromagnetically "thin." We then show through a multiobjective optimization that a low-profile coplanar sense coil geometry, in other words "flat," has a comparable performance to other optimized configurations.

A. Eddy Current Losses Dissipated in Open-Circuited Sense Coils
The sense coils are supplemental to a WPT charging system and should neither impact the efficiency nor perturb the electromagnetic fields. The sense coils are open-circuited, and hence do not carry terminal current, so there is no ohmic loss. A potential loss mechanism may be the eddy currents induced by the time-varying magnetic fields generated by the Tx and Rx coil currents; however, these losses are negligible when the sense coils are very thin; 50-AWG (0.025-mm diameter) coaxial cable is available commercially.
Eddy current losses in the sense coils were analyzed through 2D axisymmetric FEM simulations with an extremely fine mesh in COMSOL. Open-circuited single-turn sense coils were placed 1 cm above the Tx coil, as shown in Fig. 21, where the Tx and Rx coils were multiturn concentric circular coils, used in Section III-C for the WPT2/Z1 class. Eddy current losses were calculated for different numbers (one to six) and diameters (32 to 50 AWG, at 6-AWG intervals) of sense coils. Fig. 22 shows the efficiency loss from eddy currents in the sense coils. The diameter of the sense coils, which scales the loss as approximately cubic, is the dominant factor; whereas, the number of sense coils scale the loss linearly. Using commercially available 50-AWG coaxial cable, the eddy current loss only contributes less than 10 −6 % efficiency loss according to the results from these high fidelity electromagnetic FEM simulations.

B. Physically Flat Sense Coils: Performance Comparisons With Other Configurations
Physically flat sense coils are unobtrusive and can be installed above the transmitter coil and below the pavement if needed. In this section, we show that coplanar sense coils have comparable performance with other configurations. We use Monte Carlo methods to evaluate three performance metrics: 1) Model matching; 2) information diversity; and 3) detectability. Pareto frontiers from a multiobjective optimization [47] of each configuration is used for comparison. The Pareto frontier is the best performance set for a particular configuration. In other words, by comparing Pareto frontiers, one compares the best cases of each configuration. Specifically, we compare coaxial sense coils that are restricted to the same plane, i.e., flat/coplanar, with those that are only restricted coaxially, but otherwise unrestricted.
A multiobjective optimization problem was formulated from the weighted sum of three penalty functions [48] minimize r, d Each penalty function (p 1 , p 2 , and p 3 ) ranges from zero to one, normalized over the Monte Carlo data from all the comparison configurations.
Together, these penalty functions represent a tradeoff between 1) errors from mismatch between the sense coils and the models used for TPM reconstruction; 2) sensitivity to measurement errors and noise in calibration; and 3) signal-to-noise ratio in the sense coil measurement. The sense coil parameters that determine the penalty functions include the sense coil radii r i and the vertical distance from the Tx coil to the sense coils d T :i for N sense coils, where i ∈ {1, . . ., N}.R min andR max represent range of sense coil radii normalized to the Tx coil radius r T .d min andd max are the sense coil positions above the Tx coil normalized to the distance between the Tx and Rx coils d T :R . λ i are the weights for the penalty functions.
The analysis and computation are tractable when eddy currents are neglected and only the principal transfer-power is used. The following subsections detail each penalty function and show the results of the comparison.
1) Model Matching: The errors in FC-TPM can be made small and insensitive to variation when the sizes and positions of the sense coils are chosen so that the deviation from the models used in reconstruction is small. p 1 penalizes model mismatch.
For example, when the mutual inductance from the Tx coil to the Rx coil varies so it is predominantly quadratic over Rx coil misalignment, placing the sense coils close to the Tx coil makes the corresponding geometric parameters also predominantly quadratic as discussed in Section IV-C. In this case, p 1 penalizes the nonquadratic deviations over Rx coil misalignment in the sense-to-Rx coil mutual inductance. The penalty function can be evaluated using Grover's expression for mutual inductance [37] with the quadratic model presented in Section IV-C1 is the mutual inductance between the sense coil and Rx coil which is calculated from the Grover; is the mutual inductance from the quadratic model; r i is the radius of the i th sense coil; d T :i is the distance between the i th sense coil and the Tx coil; x is the Rx coil misalignment; and · 2 is the l 2 [0, x max ] norm over a closed interval and a sampling of the continuous functional M R:i (·, ·, x) and its corresponding discrete data sequence M R:i (·, ·, x[n]).
2) Information Diversity: Overlapping information in the least-squares minimization of the data matrix of sense coil voltages W, defined in (53), results in poor matrix conditioning and consequently sensitivity to noise and measurement error.
In the calibration of FC-TPM, least-squares minimization is performed, where W is inverted in (57). When the sense coil voltages are independent, the calibration is robust to measurement noise. Hence, sense coils are chosen and arranged to minimize overlapping information in the voltages.
Mutual inductance can be used as a proxy for information overlap among sense coil voltages because mutual inductance is a measure of the shared magnetic flux between two sense coils and hence information. Large mutual inductance between two sense coils may physically mean that they are similar in size and/or proximal. p 2 penalizes information similarity. It is formulated by normalizing and taking the logarithm of the sum of squares of the mutual inductances between sense coil pairs. p 2 ranges from from 0 to 1: "0" indicates that all pairs of sense coils have maximum independence given the optimization constraints; "1" indicates the worst case among all sense coil configurations in all the comparison cases where M i:j (r i , d T :i , r j , d T :j ) is the mutual inductance between the i th and j th sense coil. The mutual inductance increases as the two sense coils are closer to each other d T :i → d T :j and have similar radii r i → r j , which penalizes the lack of information diversity.

3) Detectability:
The detectability is the ability of a particular FC-TPM configuration to resolve a change in the transferpower from a change in either the Tx or Rx coil current. This is equivalent to the signal-to-noise ratio (SNR) of the different sense coil voltages from the change in the Tx/Rx coil current to the instrumentation noise referred to the sense-coil voltage. The mutual inductances between the sense coils and the Tx coil M T :i or Rx coil M R:i , determine the sense coil voltages from the Tx and Rx coil currents. p 3 penalizes poor SNR in the sense coil configuration; we assume that a change in Tx current is just as likely as a change in Rx current. where Note that M T :i (r i , d T :i ) is the mutual inductance between the Tx coil and the i th sense coil; M R:i (r i , d T :i ) is the mutual inductance between the Rx coil and the ith sense coil. We choose the Tx, Rx, and sense coils to be aligned for tractability. The total SNR is the harmonic mean of the SNR of each sense coil. p 3 penalizes overly large-or small-sized sense coils, or positions which are far from both the Tx and Rx coils.

4) Comparison Results
: Good sense coils have: 1) small quadratic approximation error for the mutual inductance over Rx coil misalignment; 2) diverse information, which minimizes the least-squares errors during calibration; and 3) good SNR in the sense coil voltages for accurate transfer-power reconstruction. It is worth noting that the three penalty functions are counteractive vis-à-vis the sense coil positions and sizes. Based on observations, 1) p 1 is smaller when the sense coils are all near the Tx coil; 2) p 2 is smaller when the sense coils are far apart and have different sizes; and 3) p 3 is smaller when the sense coils are of similar size to and in joint proximity to the Tx and Rx coils. The performance metrics of coplanar sense coil configurations are compared to other sense coil geometries with fewer restrictions for N = 6 sense coils. Design classes such as sense coil configurations should be compared by their optimal designs on Pareto frontiers. The class of coplanar sense coil configurations (i) is topologically defined by the constraints that the sense coils share the same axis (coaxial) and lie on the same plane Sense coil configuration class (ii) is topologically constrained only to be coaxial. A numerical comparison was performed using Monte Carlo methods together with Manhattan sampling 14 using identical Tx 14 The simulation uses the Manhattan sampling [49]. The search space ) is uniformly partitioned into 3 × 3 rectangular regions. These 9 rectangular regions are exhaustively travelled. The sampling follows the uniform distribution inside each rectangular region. Fig. 23. Comparison results between coaxial-coplanar and coaxialunrestricted sense coils. The yellow shows the Pareto frontier (optimal points) when sense coils are restricted to coplanar configurations. The blue shows the Pareto frontier (optimal points) when there is no restriction on the coaxial sense coil placement. The gray shows some nonoptimal sense coil geometries. and Rx coil radii (r T = r R = 22.5 cm) separated by d T :R = 20 cm, which corresponds to the hardware in Section VI. The addition geometric constraints areR min = 0.44,R max = 1.78,d min = 0.05, andd max = 0.95. This search space is large enough to cover the sense-coil geometries which are physically realizable. 15 We swept λ 2 and λ 3 , and solve (74) to obtain the optimal points r * i (λ 2 , λ 3 ) and d * T :i (λ 2 , λ 3 ) on the Pareto frontiers. The optimal points for the two sense coil configuration classes, shown in Fig. 23, form surfaces which are the Pareto frontiers for each of the multiobjective optimizations. The hollow yellow circles correspond to the Pareto frontier for coplanar sense coil configurations (i); the solid blue circles correspond to less restrictive sense coil configurations (ii); and the solid gray circles correspond to nonoptimal points. These are an illustrative sampling of the 4 billion points tested. The Pareto frontiers largely overlap, which indicates that coplanar sense coils have comparable performance to less restrictive sense coil geometries.

VI. HARDWARE RESULTS
We demonstrated FC-TPM in hardware with a 1-kW wireless charging system that operates at 85 kHz using 270 strands of 38-AWG (0.101-mm diameter) litz wire for the Tx and Rx coils. The Tx and Rx coils are each ten-turn solenoids, whose diameter is 45 cm with an air gap (d T :R ) of 20 cm between the coils. Each of the FC-TPM sense coils are single-turn, open-circuited, and placed on the same plane 2.5 cm above the Tx coil (d T :i ), as shown in Fig. 24. A very thin coaxial cable, 42 AWG (0.06335mm outer diameter), was used for each single-turn sense coil winding. The coaxial cable was configured so the outer braid acted as an electrostatic shield from the high voltage Tx and Rx windings 16 [50]. 15 To verify, sense coil geometries outside of this search space were sampled in Monte Carlo simulations; the results do not belong to the Pareto frontier. 16 Only one terminal of the shield was grounded to preclude a shorted turn.    Table III presents the specifications of the Tx, Rx, and sense coils. 17 The Tx and Rx coil were driven by identical current-mode class D (CMCD) converters [51]- [53]; the power level from the Tx to the Rx coil was adjusted by changing the phase angle of the gate signals, hence changing the corresponding WPT coil voltages and currents [51]. The dc output of the receiver was recirculated to the input of the transmitter, with a single dc power supply V dc holding the voltage of the shared dc node fixed while supplying the power loss. The CMCD WPT circuit configuration is shown in Fig. 25; one of the CMCD printed circuit boards is shown in Fig. 26. 1.2-kV SiC MOSFETs were used to block the 1-kV drain voltages. Table IV lists the components specifications. The resonant capacitors were chosen to carry the nearly 12-A rms current. Fig. 27 shows the Tx and Rx coil voltages and  currents that demonstrate kW-level WPT, driven by two CMCD converters. Note that V TX and −I RX (current into the receiver) are nearly in phase to deliver real power from the transmitter to the receiver.
The FC-TPM system was built as shown in Fig. 28. Current transformers (Pearson Model 110) were used to measure the Tx and Rx coil currents. The Tx and Rx coil current, and sense coil voltage data were each 16 megasample recordings on an Elsys TraNET 204E with a TCPE-8016-4S data acquisition system at 20 Msamples/s, 16-bit resolution; Fig. 29 shows the recorded coil currents and sense coil voltages.

A. FC-TPM With Two Sense Coils
We show FC-TPM can be demonstrated in hardware by confirming that the transfer-power can be accurately determined by using sense coil voltages together with calibrated geometric parameters. Using aligned and stationary Tx and Rx coils, we examined the accuracy from using only two sense coils (22.5 and 17.5-cm radii).
Only one single geometric parameter (42) needed to be calibrated using the least-squares minimization in Section IV-D performed over load. The data space spanned a variation in transfer-power by sweeping the phase difference between Tx and Rx coil currents at constant amplitude. 18 The reference standard transfer-power was measured simultaneously as described in Appendix E.
LOOCV was used to validate the accuracy of FC-TPM; the geometric parameter was determined by the calibration set (9 data points), the transfer-power of the validation set (1 data point) was determined from (41), and the reconstruction errors in (58) were calculated. The FC-TPM errors are shown in Fig. 30(b). The hardware errors ranged from -0.009% to 0.015%.
By using the appropriate litz wire in our WPT coils, eddy currents can be made negligible. In Fig. 30, two sense coils (22.5 and 17.5-cm radii) were used for the FC-TPM of aligned WPT coils described above. Calibrated principal transfer-power using (40) is shown for solid wire in COMSOL FEM simulation in Fig. 30(a) and litz wire in hardware in Fig. 30(b). From Section IV-E, using two sense coils for FC-TPM when there are eddy current losses in the WPT coils result in nonnegligible errors. This is manifested in the solid wire in Fig. 30(a) as a systematic error with increasing transfer-power; this systematic error can be interpreted as the error from the excess loss imposed by the additional eddy current winding in the transformer model in Section IV-E1. Fig. 30(b) shows errors that can be attributed in part to measurement and reference standard calibration that includes sensor error, noise, and digital quantization.  The transfer-power is also compared in Fig. 31 to the input and output dc power to highlight and demonstrate the principle that measuring transfer-power disaggregates the Tx and Rx losses to enable fair metering. FC-TPM was also demonstrated over a  wide range of power levels as illustrated in Fig. 32, where the error is less than 0.1% (-0.011% to 0.096%) from 60 W to 1 kW.

B. FC-TPM Over Misalignment
We demonstrated FC-TPM over Rx coil misalignment using six sense coils (radii from 20 to 22.5 cm at 0.5-cm intervals). The Rx coil was misaligned 19 by up to 10 cm (6 data points: 0 to 10 cm at 2-cm intervals), as shown in Fig. 33. The data matrix W and vector p spans six misalignment values and ten Tx-Rx coil current phase differences to calibrate the geometric parameters α ij according to (57). The percentage FC-TPM errors at each validation data point were calculated with (58) and plotted in Fig. 34. The error bars represent the range of errors at each misalignment point. The errors ranged from -0.087% to 0.07% and were very nearly consistent, demonstrating accurate FC-TPM over misalignment. It is worth noting that the explicit measurement of misalignment was not needed either for calibration or transfer-power estimation because the sense coil voltages and the constant geometric parameters encapsulate all the necessary information. The transfer-power is compared to the input and output dc power over misalignment in Table V and Fig. 35.
The dependence of FC-TPM errors to different numbers (from two to six) of sense coils were investigated. Fig. 36 shows the    worst-case absolute error percentage of FC-TPM at 1 kW, as defined in (61), of ten data point variations in Tx-Rx coil current phase difference at each misalignment point. The greater the number of sense coils, the smaller the FC-TPM errors, which is expected because more information is available to determine the transfer-power over misalignment.

C. Performance Comparison to Coaxially Stacked Sense Coils
We replaced the coplanar sense coils with coaxially stacked sense coils, as illustrated in Fig. 37, to show the comparable performance of FC-TPM using the two different sense coil configurations. First, we examined the accuracy of using two 22.5-cm radii sense coils, which are coaxially stacked 3.5 and 7.5 cm above the Tx coil, respectively. A total of ten data points using different equivalent output load resistances were validated, and Fig. 38 shows the hardware results; the FC-TPM errors ranged from −0.011% to 0.009%, which are comparable to the errors from using coplanar sense coils in Fig. 30(b).
We also demonstrated FC-TPM over Rx coil misalignment using six coaxially stacked sense coils (vertical distances above the Tx coil from 3 to 5.5 cm at 0.5-cm intervals), as shown in Fig. 37. Fig. 39 shows the percentage error, where the error bars represent the range of error at each misalignment point. The errors ranged from -0.082% to 0.071%, which are comparable to the errors from using coplanar sense coils in Fig. 34.

D. Future Implementation
Sense coils are the primary sensors, single-turn and opencircuited; conductors can be direct-printed on an insulator like FR4, and thus, inherently inexpensive and straightforward to manufacture. Commercial off-the-shelf analog-to-digital converters can be used to measure the sense coil voltages, and a fieldprogrammable gate array can process the data. A digital signal processing microcontroller can control the measurements and communicate with other devices. The hardware cost is expected to be similar to typical measurement systems [22], [54]- [56] including those planned for metering in the NIST-sponsored U.S. National Work Group (USNWG) on Measuring Systems for Electric Vehicle Fueling and Submetering (EVF&S) [57]. Furthermore, the noncontact method of FC-TPM has an additional cost savings in that the high voltage safety and insulation that is required for typical terminal voltage and power measurements is not needed.

VII. CONCLUSION
In this article, FC-TPM was introduced, analyzed, and demonstrated. Implementation on a 1-kW WPT hardware system shows better than 0.1% accuracy over a wide range of transfer-power, confirming fair and accurate metering of WPT, which to our knowledge, is a first among methods in EV charging.
Transfer-power was shown to be fundamentally derived from the Poynting vector, which is the power purely dispensed from the Tx to the Rx coil through the intervening space. Transferpower is profoundly different from the black box notion of a coil's electrical terminal power, which contains commingled losses from the Tx and Rx coils. Transfer-power ensures fair metering by attributing losses equitably between the charging station (Tx) and electric vehicle (Rx) based on the location of the heat dissipation as a demarcation. Economic decisions by stakeholders will require fine-grained information about efficiency and energy consumption. This is evidenced by growing efforts to standardize the accurate metering of EV charging. Ultimately, this will incentivize a continuing investment in technology and innovation.

APPENDIX A DERIVATION OF EDDY CURRENT LOSS IN THE RX COIL
The eddy currents in the coils can be modeled as a transformer winding [33], [34]. In this Appendix, the eddy current in the Rx coil I r , created by the Tx coil current I T , is modeled as a winding to represent the eddy current loss in terms of the Tx coil current and effective resistance R T :r . The equivalent circuit is shown in Fig. 40, resulting in the expression −jωL r I r − jωM T :r I T = (R r + jXr)I r where M T :r is the mutual inductance between the Tx coil and Rx coil's eddy current winding; R r and X r are the equivalent resistance and reactance in the winding. From (82), the eddy current I r can be represented in terms of the Tx current I T Transfer-power between two coils was defined in (12). The eddy current losses P Loss,eddy(Rx) can also be represented as a transfer-power P Loss,eddy(Rx) = Re jωM T :r I r I * T .
Note that the eddy current loss is the power that is transferred from the Tx coil to the eddy winding. The eddy current loss can also be found from (82) where The eddy current loss is represented in terms of the Tx coil current I T and the effective resistance R T :r . The Tx coil terminal-voltages V T can be derived using (15), (83), (87), and (88) Similarly, the Rx coil terminal voltage V R can represented by

APPENDIX B EXTRACTION OF WINDING LOSS MODEL FROM FEM SIMULATIONS
Two-dimensional axisymmetric simulations were performed with the Tx and Rx coils driven by current sources. Although in practice, litz wire is typical, solid wire was used in these analyses as worst-case examples. The Tx coil currents were chosen to be the maximum current for each power class, and the Rx coil currents were selected to satisfy the maximum power level for the class, both specified in [35]. Winding losses can be determined from terminal measurements using different setups, where the effective resistances and geometric factors can be obtained as follows.
(a) R T and R R : The effective winding resistance R T of the Tx coil can be calculated when the Rx coil is removed. The input power P Tx is Re{I * T I T }R T , which is the ohmic loss in the Tx coil. The winding resistance R T can through the current I T . Similarly, the winding resistance R R of the Rx coil can also be calculated when the Tx coil is removed.
Note that R T was obtained previously; so R T :r can be calculated Similarly, R R:t can be obtained when the Rx is driven by I R and the Tx coil is open-circuited (I T = 0) c) γ T , γ R : γ T , and γ R can be obtained when both the Tx and Rx currents are in phase, resulting in zero principal transfer-power (P Tx:Rx = 0), as discussed in Section II-B. γ T,R can be extracted from the the input and output power in (17) and (18) Table VI presents the coil specifications for different power and air gap classes. Fig. 41 shows an example with 7.7-kW output power when the Tx-to-Rx coil air gap is 100 mm, which emulates the WPT2/Z1 class; 2.5-mm radius 20 wire was used, where the Tx coil and Rx coil diameters were 650 and 250 mm, respectively.

APPENDIX C QUADRATIC APPROXIMATION FOR THE MUTUAL INDUCTANCE
OVER LATERAL MISALIGNMENT From Grover [37], the mutual inductance (M T :R ) of two coils consisting of circular filaments, which are laterally misaligned by x is 20 The Tx and Rx wire radii were identically 2.5 mm for all classes, except for the Rx wire radii for the WPT1/Z1 (1.57 mm) and WPT1/Z2 (2.02 mm). Power and Z-class are found in [35]. r T,R are the radii of the Tx and Rx coils. d T :R is the air gap between the Tx and Rx coils. The unit of the length is mm. N T and N R are the number of turns for the Tx and Rx coils, respectively. I T and I R are rms values. All the phase angle differences between the Tx and Rx coil currents are maintained at 90 • . The unit of the effective resistances R T , R R , R T :r , R R:t , γ R R T , and γ T R R is Ohms (Ω). r T and r R are the radii of the coils, d T :R is the vertical distance between two coils, x is the lateral misalignment, and k parameterizes the elliptic integrals. We assume that the normalized misalignment γ x/r R 1. This enables us to approximate the mutual inductance M T :R by a polynomial function of the normalized misalignment γ through Taylor expansion.
We first express M T :R as (100) The Taylor expansion of A in (100) can be expressed as where μ 1 = 1, μ 2 = 1 2 cos ϕ, μ 3 = 9 8 The Taylor expansion of B in (100) can be performed in three steps. The first step expands 2Ψ(k)/(πk) as a function of k, which is illustrated in the first layer of Fig. 42 2Ψ(k) The following two equations are useful for deriving (103): The second step expands k as a function of V , which is illustrated in the second layer of Fig. 42 k (113) Fig. 43. Expression for the mutual inductance of circular filaments in Grover can be well approximated by a quadratic function when the misalignment distance is not too large relative to the radii of the coils.
The third step expands V as a function of γ, which is illustrated in the third layer of Fig. 42.
From (100), (101), and (119), we can extract the function that is a quadratic approximation M T :R . Fig. 43 shows that the expression for mutual inductance in Grover can be well approximated by a quadratic function.

APPENDIX D DERIVATION OF FC-TPM FOR LOSSY COILS
The Tx, Rx, and eddy winding currents in (63) The transfer-power can thus be represented as a linear combination of pairwise-products of sense coil voltages in (65) where If there are more than four sense coils, we can choose four sense coil voltages at a time from the total of N sense coils, from which a total of N C 4 different formulations of (125) are constructed to determine the transfer-power. The transfer-power, therefore, can be determined by (65) for N > 4 sense coils, when all the formulations are summed and divided by N w = N C 4 , resulting in a linear combination of the all pairwise-product of two sense coil voltages out of N sense coils where Q = (i, j) ∈ N 2 i ≤ N, j ≤ N, and i < j (129) G ij = K ⊂ {1, . . ., N w } k ∈ K and i, j ∈ s k (131) Note that when T is a set of N sense coils, T = {1, 2, . . . , N}, S is a collection with lexical ordering of all subsets of T that consists of combinations of four sense coils, where n(S) = N w .

APPENDIX E REFERENCE STANDARD TRANSFER-POWER FOR THE CALIBRATION OF FC-TPM
Geometric constants that relate the sense coils' voltages to transfer-power need to be calibrated accurately. FC-TPM requires its geometric parameters α ij from (52) to be calibrated for metering. As shown from (53) to (57), transfer-power should be known for the calibration of α ij ; an independent measurement of transfer-power is needed for the reference standard transferpower.
Equation (12) shows that principal transfer-power is determined by the mutual reactance ωM T :R from the Tx coil to the Rx coil, and the Tx and Rx coil currents I T , I R . We decompose the mutual reactance ωM R:T to 1) the standard mutual reactance ωM R:T that can be measured by only using magnitude of the open-circuited Rx coil voltage V R and Tx coil current I T , and 2) γ T R R , which accounts for the electromagnetic coupling from the Tx coil current to the Rx coil, as presented in Section III-A2. The transfer-power is then where ωM R:T is derived from the open-circuited Rx coil voltage V R , which can be derived from (90) in Appendix A with zero Rx coil current and the reference standard mutual reactance ωM R:T is defined as the magnitude of ratio of the V R to the I T The reference standard transfer-powerP Transfer is then defined by ωM R:TP

Transfer
Re jωM R:T I R I * T . Note that when γ T R R is negligible (γ T R R ωM R:T ), the true transfer-power P Transfer in (133) is accurately approximated by P Transfer , where the percentage error M is M = P Transfer − P Transfer P Transfer × 100 (%) The error M is particularly small especially for the typically loosely coupled air-core WPT coils that have negligible γ T R R . 2-D axisymmetric FEM simulations were performed to verify the negligible error M of reference standard transfer-power for WPT coils, which was used in the hardware demonstration. Fig. 44 illustrates the flow chart for calibration and subsequent transfer-power measurement. As a worst-case simulation, 2.5mm diameter solid copper wires were used to make 45-cm diameter single-turn circular coils; ten of these circular coils were stacked to emulate the solenoidal Tx and Rx coils, where the solenoids' center-to-center air gap was 20 cm. The error M is only 6.7 × 10 −7 % when ωM R:T is 6.967 Ω.