### I. Introduction

### II. Previous Design Flow for a WPT System

*ω*

_{o}and AC input voltage

*V*

*for specific load conditions can also be determined by calculations or simulations.*

_{1}*ω*

_{o}or AC input voltage

*V*

_{1}can also be adjusted. The design parameters of the power source, such as the DC input voltage, duty ratio, and gate-driving control circuits, are also adjusted to obtain the desired AC voltage level. As a result, the various design parameters in the WPT system, including coils, compensation networks, the power source, and even the rectifier, should all be tuned and debugged in the final step. Hence, finding the root causes of the discrepancies between the design targets and the achieved experimental results is quite challenging, and a high level of engineering experience and iteration techniques are required to achieve the desired performance of a WPT system.

### III. Proposed Design Flow for a WPT System

### 1. Formulation of WPT Performance Parameters with Impedance Parameters in a Frequency Domain

*Z*)–parameters herein. Fig. 2 shows the overall WPT system from a DC power source to an output load. The DC voltage

*V*

*is supplied by a DC voltage supply or an AC-DC converter with a DC-link capacitor,*

_{dc}*C*

*.*

_{dc}*Z*

*represents the impedance of the rectifier and output load. Since the frequency domain formulations are intended to analyze the AC-AC characteristics of the WPT system, Port 1 is located at the AC input stage after the DC-AC inverter, and Port 2 is located at the AC output stage before the rectifier, as depicted in Fig. 2. Also, in the frequency domain analysis, the DC-AC inverter side should be disconnected, so that*

_{L}*V*

_{1}and

*I*

_{1}at Port 1 represent the AC driving voltage and current. In the conventional frequency-domain characterization method, only the WPT coils and compensation networks are characterized as a two-port network. The load is separately characterized by additional measurements and combined with the two-port network to investigate the whole performance of the WPT system. However, the separate measurements should increase the overall errors in predicting the performance parameters. In this paper, the performance parameters are derived by a measurement with the load also connected, which can include all parasitic stray inductances or contact resistances in the PCB layout and assembly. This can significantly increase the accuracy in the frequency-domain characterization.

*P*

*, imaginary power*

_{in}*Q*

*, and apparent power*

_{in}*S*

*are expressed as:*

_{in}*Y*

*|*

_{in}

_{I}_{2=0}represents the input admittance at Port 1 when

*I*

_{2}is zero, which is identical to the inverse of the

*Z*

_{11}parameter.

*V*

_{1}is also a complex number representing the magnitude and phase of the input sinusoidal voltage. Actually, the AC voltage from the DC-AC inverter output is usually a rectangular waveform with the amplitude of

*V*

*. Therefore,*

_{dc}*V*

_{1}corresponds to the fundamental frequency component of the rectangular waveform, if the power transfer of the harmonic components is negligible.

*Z*-parameters as:

*I*

*is the current through the load impedance*

_{L}*Z*

*, as illustrated in Fig. 2.*

_{L}*I*

_{2}is zero in all the expressions, and the Port 2 termination of the measuring instrument has no effect.

*Z*

*, is required for (5) and (6). If the load impedance is purely resistive as*

_{L}*R*

*, the equations are simplified as:*

_{L}*G*

*can also be obtained from the*

_{v}*Z-*parameters as:

*Z*-parameters in a frequency domain. The expressions are similar to those in [17], but do not require any circuit models in advance. In the proposed experimental optimization method, the

*Z*-parameters are experimentally obtained from the

*S*-parameters measured by a vector network analyzer (VNA). Since the expressions are not restricted by a load impedance as well as a source impedance, the WPT performance parameters obtained from (1)–(9) in a frequency domain are compatible with those of the final WPT system in a time domain. The WPT performances extracted experimentally using the proposed method should be much more accurate than any calculations or full-wave simulations, since the results are obtained from measurements with the actual WPT system being designed.

### 2. Proposed Systematic Design Flow

*ω*

_{o}and

*V*

_{1}can be tuned and optimized in advance of manufacturing the power driving and load circuits. When the power transfer characteristics of the WPT system expected from the manufactured coils and compensation networks agree well with the design targets, the power source, rectifier, and control circuits are manufactured at Step 4. After finishing all manufacturing, the actual power transfer experiments in a time domain are finally conducted with all the manufactured parts. If the overall operations in the time domain meet the design target, the design and implementation of the WPT system has been done successfully. Otherwise, the remaining design factors such as the input DC voltage and dead-time (or duty ratio) are adjusted to consider parasitic loss at the power source and rectifier stages, since the manufactured AC-AC stage of the WPT system is already validated with the frequency domain measurement in Step 3′.

### IV. Example of the WPT System Design

### 1. Design for WPT Coils

*n-*turn square spiral coil can be obtained as:

*M*

_{S}_{1,}

*is mutual inductance between two coaxial single-turn coils, while the*

_{ij}*L*

_{S}_{1,}

*is self-inductance of the coils. Additionally, the mutual inductance between an*

_{i}*n*-turn coil and

*m*-turn spiral coils with distance

*d*is obtained as:

### 2. Design of Compensation Networks

*LCC*topology, are also applicable to practical WPT systems. The power transfer characteristics of different compensation topologies are well summarized in [9], and a proper topology can be selected to minimize the VA rating and achieve constant voltage or current.

*Z*

*. For the symmetric primary and secondary coils,*

_{L}*n*

_{1}=

*n*

_{2}, the leakage inductance

*L*

*is calculated as (*

_{leak}*L*

*−*

_{self}*M*), and the magnetizing inductance

*L*

*is identical to the mutual inductance*

_{m}*M*[25].

*R*

*implies the parasitic resistance from each coil and compensation network, including the equivalent series resistance (ESR) of the capacitors. The impedance parameters,*

_{loss}*Z*

_{11}and

*Z*

_{21}, can be derived from the circuit model as:

*Z*

*=*

_{M}*j*

*ω*

*L*

*,*

_{M}*Z*

*=*

_{leak}*j*

*ω*

*L*

*, and*

_{leak}*Z*

*= 1/(*

_{c}*j*

*ω*

*C*).

*R*

*is usually much smaller than the load impedance, and the summation of*

_{loss}*R*

*and*

_{loss}*Z*

*in (12) and (13) can be simply approximated to*

_{L}*Z*

*.*

_{L}*ω*

_{o,1}, which implies (

*Z*

*+*

_{M}*Z*

*) +*

_{leak}*Z*

*= 0,*

_{c}*ω*

_{o,1}can be written as:

*ω*

_{o,2}, which implies

*Z*

*+*

_{leak}*Z*

*= 0,*

_{c}*ω*

_{o,2}is written as:

*ω*

_{o}_{,1}and the leakage inductance compensation frequency

*ω*

_{o}_{,2}are within the target frequency range for the manufactured coils. The self-inductance compensation frequency

*ω*

_{o}_{,1}was obtained as 2π · 43.4 krad/s regardless of the distance between the coils. The leakage inductance compensation frequency

*ω*

*was obtained as 2π·49.5 krad/s for a coil distance of 150 mm, but the frequency decreases with increased coil distance or any misalignments.*

_{o,2}*Z-*parameters. The approximated expressions for the two compensation conditions are summarized in Table 3. It is assumed that

*R*

*is much smaller than the magnitudes of*

_{loss}*Z*

*and*

_{M}*Z*

*.*

_{L}*R*

*and*

_{L}*X*

*represent the real and imaginary parts of the load impedance, respectively.*

_{L}*ω*

_{o,1}, if the imaginary part of the load impedance is negligible. Therefore, the output power is maximized at frequency

*ω*

_{o,1}regardless of the mutual inductance and load variation. However, one weakness of this compensation strategy is that input power, output power, and voltage gain can be significantly changed, depending on the distance between coils and load variation, which causes difficulty in the control circuit design of the power source or rectifier. Nevertheless, the leakage inductance compensation strategy can achieve the unity voltage gain at operating frequency

*ω*

_{o,2}regardless of the load variation. However, the operating frequency

*ω*

_{o,2}itself is changed by the distance and misalignment between two coils, since the leakage inductance depends on mutual inductance. Therefore, another control technique is needed to adjust the operating frequency. The expressions of the power transfer efficiency at both

*ω*

_{o,1}and

*ω*

_{o,2}are identical according to Table 3; yet, the efficiency variations around the frequencies

*ω*

_{o,1}and

*ω*

_{o,2}differ from each other due to the different assumptions for

*R*

*, which will be shown in the next section with the experimental results.*

_{loss}*ω*

_{o,1}in the WPT design herein. According to the analytical calculation results, the |

*V*

_{1}| should be 30.71 to achieve 100 W output power at a 40 Ω load for the 150 mm air gap between the coils.

### V. Experimental Setup and Validations

*S*-parameters of the AC stages in the WPT system have been measured with the load resistor connected, but the DC-AC power source disconnected, as depicted in Fig. 2. The measured

*S*-parameters are then converted to

*Z-*parameters, and

*Z*

_{11}and

*Z*

_{21}are substituted to (1)–(6) and (9). The impedance of the cement resistors, including large parasitic inductances, were separately characterized by the VNA measurements to achieve high accuracy, rather than directly utilizing the Eqs. (7) and (8). As mentioned in Section III,

*V*

_{1}corresponds to the fundamental frequency component of the DC-AC power source. When the power source is implemented with the full-bridge inverter and the first harmonic approximation is valid, the magnitude of

*V*

_{1}is equal to 4/π times that of

*V*

*. For the power source as the half-bridge inverter, it is 2/π times that of*

_{dc}*V*

*.*

_{dc}*R*

*and switching loss in the inverter, the actual output power should be lower than the expected value. The power analyzer is used to simultaneously measure the input and output powers of the AC-AC stage.*

_{loss}*ω*

_{o,1}. Additionally, the unity gain is achieved at 50.0 kHz, which is also slightly different from the operating frequency for leakage inductance compensation

*ω*

_{o,2}. The power efficiencies at 20 kHz, extracted using the two methods, have relatively large discrepancies due to the high frequency distortion. However, except for the input power factor and the power transfer efficiency at the 20 kHz operation, the discrepancies at other operating frequencies are acceptable by considering the tolerance of the components and the distance between two coils.

*ω*

_{o,2}decreases down to 46 kHz. The efficiencies in all the frequency ranges change by similar amounts, which is a different trend than in Fig. 11.

_{o,1}

*M*)

^{2}/

*R*

*from equations (12)–(14). Thus, the input impedance at the self-inductance compensation frequency decreases when the load resistance increases or mutual inductance decreases; then, the input power increases if the input voltage is constant. However, when the power transfer efficiency through the WPT system does not decrease that much, the output transferred power also significantly increases.*

_{L}*V*

_{1}| = 39.47) to achieve the design target of 100 W output power at this operating frequency. The final performance parameters at the operating condition are summarized in Table 4. The final performance parameters from the time domain measurements are compared with those from the proposed method using VNA, when the load resistance is 40 Ω and the distance is 150 mm. The final measurement results in the time domain agree well with the estimation by the proposed method using frequency domain measurements as well as the initial design targets.