### I. Introduction

*Z*

*), which takes into account the nonlinear behavior of the rectifier circuit, including the diode components. Although the equivalent circuit model with nonlinear components can be analyzed in the time domain using a circuit simulation tool, such as SPICE, the non-intuitive time-domain simulation results cannot interpret the system response according to the adaptively changing operating frequency. Therefore, a study proposed a linearized equivalent AC resistance model with the rectifier circuit of the receiver [7], while several researchers have modeled and analyzed the receiver and rectifier using the same model [8, 9]. However, as the model assumes that the forward voltage of the rectifier diode is zero, a considerably large error occurs when designing mobile products that use a relatively small receiving voltage of approximately 5 V. A previous study [10] included the forward voltage of the diode in the model to reduce the error; however, the disadvantage was the requirement for additional experiments for modeling the rectifier through measurements.*

_{eq}### II. Proposed Iterative Method for Modeling a WPT System with Nonlinearity

*v*

*(*

_{s}*t*), with an amplitude of

*V*

*, period*

_{s}*T*, and duty cycle

*D*. Based on the Fourier series expansion, the voltage source can be expressed as indicated in Eq. (1), where

*h*denotes the harmonic number of the order. The square wave source is transferred to the rectifier circuit via the resonant network, resulting in DC voltage at the load.

*R*

*,*

_{T}*C*

*,*

_{T}*L*

*and*

_{T}*R*

*,*

_{R}*C*

*,*

_{R}*L*

*denote the equivalent R, C, L components of the transmitting and receiving coils, respectively. The transfer function of the resonant network with a linear load*

_{R}*R*

*can be modeled as indicated in Eq. (2).*

_{L}*D*

_{1}and

*D*

_{2}conduct in series, allowing the flow of input current

*i*

*, whereas diodes*

_{b}*D*

_{3}and

*D*

_{4}are blocked; this behavior is reversed in the negative half-cycle. The rectified current,

*i*

*, is filtered by the smoothing capacitor,*

_{d}*C*

*, connected in parallel with the load,*

_{L}*R*

*, generating the DC output of the full-bridge rectifier circuit. The DC output*

_{L}*I*

*can be obtained by integrating the rectified current, as indicated in Eq. (3).*

_{o}*v̂*, is clamped by the DC output of the rectifier, owing to a voltage increase that is twice the forward voltage of the diodes (

_{b}*v*

*). Hence, the amplitude of the fundamental component of the rectifier input voltage (*

_{f}*v̂*

*(*

_{s}*t*).

*V*

*, can be expressed as the amplitude of the square wave source, the transfer function of the resonant network, and the forward voltage of the diodes, as indicated in Eq. (7). The input impedance of the rectifier circuit (*

_{o}*Z*

*), which is defined by the amplitude of the fundamental components of the voltage and current at the rectifier in the fundamental AC analysis, can be obtained by applying Eq. (4) to Eq. (5), as indicated in Eq. (8).*

_{eq}*I*

*denotes the saturation current,*

_{S}*N*indicates the ideality factor, and

*V*

*represents the thermal voltage, as indicated in Eq. (9). Considering that the forward voltage of the diodes (*

_{T}*v*

*) is dynamically changed by the diode current,*

_{f}*i*

*, which is determined using the output DC voltage*

_{d}*V*

*, it can be updated by applying Eq. (4) to Eq. (9).*

_{o}*v*

*(0), and the output voltage,*

_{f}*V*

*(0). Based on these initial conditions, the input impedance of the rectifier circuit,*

_{o}*Z*

*, is obtained, and the output voltage,*

_{eq}*V*

*(*

_{o}*i*), is determined with respect to the transfer function of the resonant network associated with

*Z*

*. The output error,*

_{eq}*V*

*, is considered the absolute value of the difference between the updated and previous output voltages. If the output error lies beyond the output tolerance range,*

_{o_err}*V*

*, the diode current is calculated using the updated output voltage. Additionally, the forward voltage of the diodes,*

_{o_tol}*v*

*(*

_{f}*i*+1), is updated, which serves as feedback for the next iteration. This forward voltage updates both the input impedance of the rectifier and the output voltage in the next recursion. The proposed method repeats the iteration until the output error falls within the output tolerance range in the converged model. Thus, the proposed method uses the dynamic update of the forward voltage of the diodes and the corresponding output voltage to determine the nonlinear output response of a WPT system, considering the convergence in the steady state.

### III. Experimental Verification

*R*

*= 20 Ω. We used Keysight MSO-X 4154A to measure*

_{L}*V*

*to maintain a constant value of 19 V while changing the system operating frequency from 120 to 210 kHz, where the WPT system operates in the inductive region of the resonant network. Conversely,*

_{S}*V*

*was measured using an Agilent U1272A digital multimeter. Table 2 summarizes the design parameters of the resonant network, which were determined considering a resonant frequency of 100 kHz. The transmitting and receiving coils with a radius of 43 mm and 44 mm, respectively, were separated by a distance of 7 mm and fabricated to achieve the self-inductances shown in Table 2.*

_{o}*v*

*(0) and*

_{f}*V*

*(0), were set to 0 V and 5 V, respectively, with an output voltage tolerance,*

_{o}*V*

*, of 10*

_{o_tol}^{−6}. Fig. 6 illustrates the convergence plot of the forward voltage of the diodes and the output voltage as the iteration repeats with an operating frequency of 150 kHz. After eight iterations, the output error attained the tolerance range, and the iteration was terminated with the converged forward voltage of the diodes and an output voltage of 0.60 V and 4.79 V, respectively.