I treat the wireless charging system of an electric vehicle as a tightly coupled resonant power converter whose efficiency is governed by the instantaneous magnetic coupling, the reflected impedance, and the equivalent load presented to the secondary winding. The central problem I address is not merely the reduction of mutual inductance under coil misalignment, but the complete chain of consequences that follows: the reflected resistance moves away from its efficient range, the input impedance phase changes, the DC/DC converter no longer presents an optimal load, and losses redistribute between the primary coil, the secondary coil, the magnetic core, and the power electronics. I build a control strategy around this chain. I first derive an impedance model that exposes the relationship between mutual inductance and reflected resistance, then map the duty cycle of a secondary-side Buck converter to an equivalent AC load, then estimate mutual inductance online from voltage and current phase information, and finally coordinate the estimation and load matching through a hierarchical controller with explicit boundary limits. I use simulation and repeated power measurements to quantify the efficiency gain, the robustness of the identification, and the loss constraints of a three-coil electric vehicle wireless charging system.
I write the analysis for an electric vehicle wireless charging system operating near an 85 kHz resonant frequency. The system is series-compensated on both sides, the primary is supplied from a 400 V DC bus, the secondary delivers a regulated 360 V to a battery load, and the rated power is 3.3 kW. The air gap is 140 mm. The lateral offset ranges from 0 mm to 120 mm. This range is representative of an electric vehicle that stops with a lateral parking error after an automatic parking maneuver. The offset is not a small perturbation. It changes mutual inductance, shifts the reflected impedance, changes the current distribution in a three-coil arrangement, and can move the converter away from its high-efficiency operating region. I therefore formulate the control problem as a joint estimation-and-matching problem rather than as a fixed-frequency tuning problem.
I define the main symbols in Table 1. The symbols are used consistently in the impedance equations, the load-mapping equations, the Kalman filter, and the loss decomposition. The table also clarifies which quantities are measured directly, which are estimated online, and which are calibrated offline.
Table 1. Symbols and roles in the electric vehicle wireless charging model.
| Symbol | Meaning | Role |
|---|---|---|
| \(R_1\) | Primary coil AC resistance | Calibrated and temperature-compensated |
| \(R_2\) | Secondary coil AC resistance | Calibrated and temperature-compensated |
| \(L_1, L_2\) | Self-inductances of primary and secondary coils | Measured offline and used in the resonance model |
| \(C_1, C_2\) | Series compensation capacitors | Selected for resonance near 85 kHz |
| \(M\) | Mutual inductance | Estimated online; primary indicator of coupling state |
| \(\omega\) | Angular frequency, \(2\pi f\) | Nominal \(f=85\,\mathrm{kHz}\) |
| \(R_{ac}\) | Equivalent AC load seen by the secondary | Mapped from the battery load and Buck duty cycle |
| \(R_{ref}\) | Reflected resistance at the primary side | Used to evaluate load matching |
| \(D\) | Buck converter duty cycle | Control variable for load matching |
| \(R_b\) | Battery-side equivalent resistance | Approximately \(39.27\,\Omega\) at 360 V and 3.3 kW |
| \(P_{loss}\) | Total loss | Sum of copper, core, switching, and harmonic losses |
| \(\eta\) | End-to-end efficiency | Primary DC input to battery DC output |
I begin with the two-coil series-compensated model. The primary input impedance is the sum of the primary coil resistance, the primary reactive term, and the impedance reflected from the secondary. When higher-order parasitic effects are ignored, the input impedance can be written as
$$Z_{in}=R_1+j\left(\omega L_1-\frac{1}{\omega C_1}\right)+\frac{\omega^2 M^2}{R_2+R_{ac}+j\left(\omega L_2-\frac{1}{\omega C_2}\right)}.$$
Near resonance, the imaginary terms of the primary and secondary self-impedances are close to zero. The reflected impedance is then predominantly resistive, and I write the reflected resistance as
$$R_{ref}=\frac{\omega^2 M^2}{R_2+R_{ac}}.$$
This expression is the first link in the control chain. If the electric vehicle stops with a lateral offset, the mutual inductance \(M\) decreases. The reflected resistance decreases with the square of \(M\). A fixed duty cycle therefore produces a reflected resistance that is too low, and the primary-side converter operates at a point where conduction loss and reactive circulation are no longer balanced. The efficiency problem is thus not simply “less coupling” but “wrong load after coupling changes.” I use this observation to justify a load-matching control loop rather than a fixed compensation design.
I express the primary current from the input voltage and the input impedance. If \(V_1\) is the fundamental primary voltage and \(I_1\) is the fundamental primary current, the phase angle between them is
$$\phi=\arg\left(Z_{in}\right)=\arctan\left(\frac{\Im\{Z_{in}\}}{\Re\{Z_{in}\}}\right).$$
When the compensation network is close to resonance, the real part of the input impedance is approximately \(R_1+R_{ref}\), and the imaginary part is a small residual detuning term. I use the phase information twice: first to detect whether the system is operating near resonance, and second to separate mutual-inductance variation from compensation detuning. This separation is important for an electric vehicle wireless charging system because a change in phase can be caused either by a coil offset or by a change in the compensation reactance. If the controller interprets both as a mutual-inductance change, it will command the wrong duty cycle.
I write the efficiency of the two-coil link in a form that exposes the loss allocation. The power delivered to the secondary side is associated with the reflected resistance, while the primary copper loss is associated with \(R_1\). A simplified link efficiency can be expressed as
$$\eta_{link}=\frac{R_{ref}}{R_1+R_{ref}}\cdot\frac{R_{ac}}{R_2+R_{ac}}.$$
This expression is not a complete converter model, but it shows the trade-off clearly. Increasing \(R_{ac}\) by reducing the duty cycle reduces the secondary current for a given power, but it also changes the reflected resistance. Decreasing \(R_{ac}\) has the opposite effect. The optimum is not at an extreme duty cycle. It lies in an interval where the reflected resistance is large enough to reduce primary circulation loss but not so large that secondary conduction loss dominates. I therefore use a partitioned load-matching law instead of a single global gain.
I now map the battery load to the AC side. The secondary uses a full-bridge rectifier followed by a continuous-conduction Buck converter. The battery-side equivalent resistance is \(R_b\). For a Buck converter with duty cycle \(D\), the DC input resistance of the converter is approximately
$$R_{dc}=\frac{R_b}{D^2}.$$
The rectifier transforms this DC resistance into an equivalent AC resistance. Using the fundamental approximation for a full-bridge rectifier, I write
$$R_{ac}=\frac{8}{\pi^2}R_{dc}=\frac{8}{\pi^2}\frac{R_b}{D^2}.$$
Combining this mapping with the reflected-resistance equation gives
$$R_{ref}=\frac{\omega^2 M^2}{R_2+\frac{8R_b}{\pi^2 D^2}}.$$
This equation is the core of the load-matching strategy. For a given \(M\), I can choose \(D\) so that \(R_{ref}\) approaches the high-efficiency interval. When \(M\) falls because of misalignment, I increase \(D\) to reduce \(R_{dc}\) and \(R_{ac}\), which raises the reflected resistance. The control action is therefore physically interpretable: the duty cycle compensates for the loss of magnetic coupling by changing the load presented to the secondary. I do not treat the duty cycle as a generic actuator. I treat it as a load-matching variable whose effect is predicted by the impedance model.
I illustrate the nominal operating point in Table 2. The rated battery load is approximately \(39.27\,\Omega\) at 360 V and 3.3 kW. At a lateral offset of 80 mm, the mutual inductance is approximately \(19.6\,\mu\mathrm{H}\). When the duty cycle is adjusted from 0.42 to 0.51, the reflected resistance moves from about \(0.61\,\Omega\) to about \(0.90\,\Omega\), which is close to the high-efficiency interval around \(0.92\,\Omega\). This calculation explains why partitioned load matching is theoretically reasonable for an electric vehicle wireless charging system. It also shows that the required duty-cycle change is modest but not negligible.
Table 2. Nominal parameters and a representative offset operating point.
| Parameter | Value | Remark |
|---|---|---|
| Rated power | 3.3 kW | Battery-side DC output |
| Operating frequency | 85 kHz | Resonant frequency |
| Primary DC input | 400 V | Regulated DC bus |
| Secondary target voltage | 360 V | Battery interface |
| Air gap | 140 mm | Nominal vertical separation |
| Lateral offset range | 0–120 mm | Representative parking error |
| Battery equivalent resistance | \(39.27\,\Omega\) | At 360 V and 3.3 kW |
| Mutual inductance at 80 mm | \(19.6\,\mu\mathrm{H}\) | Reduced coupling |
| Duty cycle before matching | 0.42 | Fixed-parameter baseline |
| Duty cycle after matching | 0.51 | Partitioned control |
| Reflected resistance before | \(\approx 0.61\,\Omega\) | Below efficient interval |
| Reflected resistance after | \(\approx 0.90\,\Omega\) | Near efficient interval |
I also examine the three-coil case because a relay coil can improve the magnetic field distribution, but it introduces additional loss. For an electric vehicle wireless charging system with a relay coil, the efficiency gain is not automatic. The relay coil carries current, and that current produces copper loss. Under offset, the relay current rises, the skin effect becomes stronger, and the harmonic content increases. I write the relay-coil loss as
$$P_{relay}=I_r^2R_{r,ac}+P_{skin,r}+P_{harm,r},$$
where \(I_r\) is the relay-coil RMS current, \(R_{r,ac}\) is its AC resistance, \(P_{skin,r}\) represents the additional skin and proximity loss, and \(P_{harm,r}\) represents the harmonic loss. The total efficiency is then
$$\eta=\frac{P_{batt}}{P_{in}}=\frac{P_{in}-\left(P_{p,cu}+P_{s,cu}+P_{r,cu}+P_{core}+P_{sw}+P_{harm}\right)}{P_{in}}.$$
This decomposition matters because a two-coil efficiency improvement cannot be transferred directly to a three-coil electric vehicle wireless charging system. The relay coil can improve coupling, but its loss can consume part of the gain. I therefore evaluate both the efficiency gain and the loss breakdown instead of reporting a single efficiency number.
My identification method uses synchronized voltage and current samples. I sample at 2 MS/s, apply a fourth-order Butterworth low-pass filter with a 300 kHz cutoff, and then apply a digital band-pass filter from 70 kHz to 100 kHz. I use a Hanning window with a length of four fundamental periods and perform a sliding discrete Fourier transform to extract the fundamental amplitude and phase. The sliding transform is attractive for this electric vehicle wireless charging application because it provides a new phase estimate at each sample update without requiring a full block recomputation. I use the phase to obtain an initial mutual-inductance estimate and then use a Kalman filter to reduce measurement noise.
For the initial estimate, I express the reflected resistance from the primary power balance. If \(P_{in}\) is the primary input power, \(I_1\) is the RMS primary current, and \(R_1\) is the primary AC resistance, then the power transferred to the reflected resistance is approximately \(P_{in}-I_1^2R_1\). I therefore write
$$R_{ref,0}=\frac{P_{in}-I_1^2R_1}{I_1^2}.$$
The initial mutual-inductance estimate is then
$$M_0=\frac{1}{\omega}\sqrt{R_{ref,0}\left(R_2+R_{ac}\right)}.$$
When the compensation network is slightly detuned, I remove the calibrated reactive component from the input impedance before computing the initial estimate. This prevents a frequency shift from being misclassified as a coupling change. The phase extraction is still used, but its role is diagnostic: it tells me whether the reactive residual is within the calibrated band. If the residual exceeds the band, I flag a compensation or frequency-tracking issue rather than adjusting the mutual-inductance estimate.
I use a second-order Kalman model for the mutual inductance. The state vector is \(x_k=[M_k,\dot M_k]^T\), where \(\dot M_k\) is the rate of change of mutual inductance. The discrete-time model is
$$x_k=F x_{k-1}+w_{k-1},\qquad F=\begin{bmatrix}1&\Delta t\\0&1\end{bmatrix},$$
where \(w_{k-1}\) is process noise. The measurement is
$$z_k=M_{0,k}+v_k,\qquad H=\begin{bmatrix}1&0\end{bmatrix},$$
where \(v_k\) is measurement noise. I apply the standard prediction and update equations:
$$\hat{x}_k^-=F\hat{x}_{k-1},$$
$$P_k^-=FP_{k-1}F^T+Q,$$
$$K_k=P_k^-H^T\left(HP_k^-H^T+R\right)^{-1},$$
$$\hat{x}_k=\hat{x}_k^-+K_k\left(z_k-H\hat{x}_k^-\right),$$
$$P_k=\left(I-K_kH\right)P_k^-.$$
The covariance matrices \(Q\) and \(R\) are tuned from repeatability measurements rather than from a single trial. I set \(Q\) to allow a plausible rate of change during a parking maneuver, but not so large that the filter follows every noise spike. I set \(R\) from the variance of the phase-based estimate under known coupling conditions. The filter output is the estimated mutual inductance used by the controller. The rate estimate is not used as a direct control variable; it is used only to detect fast transients and to hold the zone decision until the estimate settles.
I divide the coupling state into four zones. The zones are defined by mutual inductance thresholds and are summarized in Table 3. The thresholds are selected so that the controller can choose a load-matching policy without continuously chasing small estimation errors. Normal coupling is at or above \(26\,\mu\mathrm{H}\). Mild offset is from 22 to \(26\,\mu\mathrm{H}\). Moderate offset is from 17 to \(22\,\mu\mathrm{H}\). Weak coupling is below \(17\,\mu\mathrm{H}\). The controller switches zones only after three consecutive estimates cross a threshold. This hysteresis prevents noise-induced chattering and protects the DC/DC converter from unnecessary duty-cycle transients.
Table 3. Coupling zones for the electric vehicle wireless charging controller.
| Zone | Mutual inductance | Control intent | Additional action |
|---|---|---|---|
| Normal coupling | \(M\geq 26\,\mu\mathrm{H}\) | Maintain high-efficiency load match | No derating |
| Mild offset | \(22\leq M<26\,\mu\mathrm{H}\) | Small duty-cycle correction | Monitor temperature |
| Moderate offset | \(17\leq M<22\,\mu\mathrm{H}\) | Partitioned load matching | Limit duty-cycle slew rate |
| Weak coupling | \(M<17\,\mu\mathrm{H}\) | Reduce output and request alignment | Limit to 50% rated power or less |
The control structure has three layers. The outer layer updates the mutual-inductance estimate and the target load every 5 ms. The middle layer uses an offline efficiency map to select the duty-cycle command. The inner layer regulates the output voltage with a 100 μs period. The inner loop is fast enough to reject bus and load disturbances, while the outer loop is slow enough to avoid interacting with the phase extraction window. I limit a single duty-cycle change to 0.005. This limit is not arbitrary. It keeps the current transient within the safe operating area of the secondary rectifier and prevents the load-matching loop from exciting the resonant network.
Table 4. Hierarchical control timing and constraints.
| Layer | Update period | Function | Constraint |
|---|---|---|---|
| Outer | 5 ms | Mutual-inductance estimate and target load | Zone hysteresis of three estimates |
| Middle | 5 ms | Efficiency-map duty-cycle selection | Duty-cycle step limit 0.005 |
| Inner | 100 μs | Output voltage regulation | Current and temperature limits |
| Protection | Event-driven | Over-current and over-temperature handling | Enter power-limit mode |
I include explicit boundary limits because an electric vehicle wireless charging system must fail safe. When the primary current exceeds 1.1 times its rated value, when the magnetic core temperature reaches 60 °C, or when the relay-coil temperature reaches 65 °C, the controller enters a power-limit mode. These thresholds are calibrated for the simulated prototype and are not presented as universal values. When the system enters the weak-coupling zone, the output is reduced to 50% of rated power or less, and a parking-alignment correction request is sent to the upper controller. I also use a 100 ms sliding average of the battery current for state-of-charge estimation, while the instantaneous current is used for over-current protection. The current ripple is kept below 2%.
I evaluate the method with an electromagnetic model built in ANSYS Maxwell and a circuit-and-control model built in MATLAB/Simulink. The electromagnetic model provides the mutual inductance and loss parameters as functions of lateral offset. The circuit model implements the resonant network, the rectifier, the Buck converter, the battery load, the phase extraction, the Kalman filter, and the hierarchical controller. I use the same operating conditions in both models: 3.3 kW rated power, 85 kHz, 400 V DC input, 360 V target output, 140 mm air gap, and 0–120 mm lateral offset. I measure power with a high-precision power analyzer and current sensors. The analyzer has 18-bit sampling precision and 15 MS/s sampling speed. The current sensor has a 50 A rating and a bandwidth from DC to 10 MHz. The voltage probe has a 200 MHz bandwidth and a \(\pm 1500\) V range. I use an LCR meter with 0.05% accuracy for static parameter measurements and a thermal camera with \(384\times288\) resolution and \(\pm2\) °C accuracy for temperature measurements.
Table 5. Measurement and simulation setup.
| Item | Specification | Use |
|---|---|---|
| Electromagnetic model | Finite-element field model | Mutual inductance and loss parameters |
| Circuit model | Resonant converter and control model | Dynamic and efficiency evaluation |
| Power analyzer | 18 bit, 15 MS/s | Input and output power |
| Current sensor | 50 A, DC–10 MHz | Primary and secondary current |
| Voltage probe | 200 MHz, \(\pm1500\) V | DC bus and rectifier waveforms |
| LCR meter | 20 Hz–2 MHz, 0.05% | Self-inductance and resistance |
| Thermal camera | \(384\times288\), \(\pm2\) °C | Core and coil temperature |
I test the mutual-inductance identification by superimposing white noise on a 0–120 mm displacement trajectory. At each signal-to-noise ratio, I run 100 Monte Carlo trials. The results are summarized in Table 6. At 20 dB, the filtered root-mean-square error is \(0.36\,\mu\mathrm{H}\), and the error standard deviation is reduced by about 60.9% relative to the unfiltered estimate. At 30 dB, the filtered RMSE is \(0.18\,\mu\mathrm{H}\), and the reduction is 61.7%. At 40 dB, the filtered RMSE is \(0.09\,\mu\mathrm{H}\), and the reduction is 59.1%. The convergence time is 18 ms at 20 dB, 14 ms at 30 dB, and 12 ms at 40 dB. Under the 20 dB condition, 95.3% of the normalized residuals lie within \(\pm2\) predicted standard deviations. I interpret this as evidence that the covariance settings are consistent with the actual estimation error and that the filter is not overconfident.
Table 6. Mutual-inductance identification under different signal-to-noise ratios.
| SNR (dB) | Pre-filter RMSE (\(\mu\)H) | Post-filter RMSE (\(\mu\)H) | Error standard deviation reduction (%) | Convergence time (ms) |
|---|---|---|---|---|
| 20 | 0.92 | 0.36 | 60.9 | 18 |
| 30 | 0.47 | 0.18 | 61.7 | 14 |
| 40 | 0.22 | 0.09 | 59.1 | 12 |
I evaluate efficiency with repeated start-stop tests at each offset. For each condition, I run 10 independent tests, wait for the system to stabilize, and then collect 1 s of power data. The results are summarized in Table 7. At 0 mm offset, the efficiency without tracking is \(93.00\%\pm0.12\%\), and the efficiency with tracking is \(93.38\%\pm0.13\%\). The gain is 0.38 percentage points, with a 95% confidence interval from 0.32 to 0.44 percentage points. At 40 mm offset, the efficiency rises from \(90.10\%\pm0.18\%\) to \(92.30\%\pm0.16\%\), a gain of about 2.20 percentage points. At 80 mm offset, the efficiency rises from \(86.50\%\pm0.21\%\) to \(90.80\%\pm0.18\%\), a gain of 4.30 percentage points. At 120 mm offset, the efficiency rises from \(80.90\%\pm0.27\%\) to \(86.20\%\pm0.24\%\), a gain of 5.30 percentage points. The response time increases from 42 ms at 0 mm to 96 ms at 120 mm. All response times are below the 200 ms control deadline of the prototype.
Table 7. Efficiency tracking results for the electric vehicle wireless charging system.
| Lateral offset (mm) | Efficiency without tracking (%) | Efficiency with tracking (%) | Gain (percentage points) | 95% CI (percentage points) | Response time (ms) |
|---|---|---|---|---|---|
| 0 | \(93.00\pm0.12\) | \(93.38\pm0.13\) | 0.38 | 0.32–0.44 | 42 |
| 40 | \(90.10\pm0.18\) | \(92.30\pm0.16\) | 2.20 | 2.07–2.33 | 55 |
| 80 | \(86.50\pm0.21\) | \(90.80\pm0.18\) | 4.30 | 4.15–4.45 | 73 |
| 120 | \(80.90\pm0.27\) | \(86.20\pm0.24\) | 5.30 | 5.12–5.48 | 96 |
I also quantify measurement uncertainty. The expanded uncertainty of efficiency at 85 kHz is 0.60 percentage points. This value is obtained by combining the uncertainties of the power analyzer, the current sensor, the voltage probe, and the repeatability of the start-stop tests. I write the combined standard uncertainty as
$$u_c(\eta)=\sqrt{\sum_i\left(c_i u_i\right)^2},$$
and the expanded uncertainty as
$$U=k\,u_c(\eta),$$
with \(k=2\). At 0 mm offset, the measured gain of 0.38 percentage points is smaller than the expanded uncertainty. I therefore do not claim a meaningful efficiency improvement at perfect alignment. At 40 mm, 80 mm, and 120 mm, the gains exceed the uncertainty boundary. The 80 mm case is the most important for an electric vehicle wireless charging system because it represents a realistic parking error: the efficiency increases from 86.5% to 90.8%, and the gain is 4.3 percentage points. The output voltage at 80 mm is \(359.6\pm0.8\) V. After 20 minutes of operation, the maximum core temperature falls from 48.6 °C to 45.2 °C. The maximum response time is 96 ms, which is below the 200 ms control limit.
I examine the three-coil loss bottleneck at 80 mm offset and 3.3 kW output. The total loss is 388 W. The relay-copper loss is 121 W, which is 31.2% of the total loss. The primary copper loss, secondary copper loss, and core loss are 16.5%, 14.9%, and 13.4% of the total loss, respectively. The third and fifth harmonics are 7.6% and 2.9% of the fundamental RMS value. The additional AC loss due to harmonics is about 19 W. This result explains why a three-coil electric vehicle wireless charging system can show a smaller efficiency gain than a two-coil system even when the coupling is improved. The relay coil adds a loss path that is sensitive to current distribution and harmonic content.
Table 8. Loss decomposition at 80 mm offset and 3.3 kW output.
| Loss component | Value (W) | Share of total loss (%) | Interpretation |
|---|---|---|---|
| Relay copper loss | 121 | 31.2 | Dominant additional loss |
| Primary copper loss | 64 | 16.5 | Primary current stress |
| Secondary copper loss | 58 | 14.9 | Secondary current stress |
| Core loss | 52 | 13.4 | Flux distribution and frequency |
| Harmonic AC loss | 19 | 4.9 | Third and fifth harmonic effects |
| Other losses | 74 | 19.1 | Switching, rectifier, and stray loss |
| Total | 388 | 100.0 | Measured and simulated balance |
I compare winding and harmonic-mitigation measures in Table 9. At 80 mm offset and 3.3 kW output, a rounded-rectangular planar spiral coil reduces AC resistance by 7.8% and hot-spot temperature by 2.1 °C compared with a right-angle rectangular winding. When I use 0.10 mm Litz wire and a 255 kHz notch branch, the third harmonic falls to 2.4%, the relay copper loss decreases by about 24 W, and the efficiency increases by 0.6 percentage points. The notch branch is not a universal solution; it must be tuned to the dominant harmonic and must not degrade the fundamental resonance. The result nevertheless shows that coil geometry and harmonic suppression must be optimized together. Improving the control alone cannot remove a loss bottleneck that is dominated by the relay-coil copper loss.
Table 9. Winding and harmonic-mitigation comparison.
| Measure | AC resistance change | Hot-spot change | Third harmonic | Efficiency effect |
|---|---|---|---|---|
| Rounded-rectangular spiral | −7.8% | −2.1 °C | No direct change | Improves thermal margin |
| 0.10 mm Litz wire | Reduced proximity loss | Lower copper hot spot | Reduced | Supports higher current density |
| 255 kHz notch branch | No fundamental penalty when tuned | Lower relay temperature | 2.4% | +0.6 percentage point |
| Relay copper loss reduction | — | Lower relay temperature | — | About −24 W loss |
I summarize the key performance indicators and constraints in Table 10. The table is intended to show the complete design logic rather than a single optimized number. The electric vehicle wireless charging controller must simultaneously satisfy efficiency, output regulation, thermal limits, response time, and alignment detection. A design that maximizes one metric while violating another is not acceptable for an electric vehicle charging interface.
Table 10. Key performance indicators and constraints.
| Indicator | Result or limit | Condition | Design implication |
|---|---|---|---|
| Efficiency at 80 mm | 90.8% | 3.3 kW, 85 kHz | 4.3 percentage point gain |
| Efficiency at 120 mm | 86.2% | 3.3 kW, 85 kHz | 5.3 percentage point gain |
| Identification RMSE | 0.36 \(\mu\)H | 20 dB SNR | Stable zone decisions |
| Response time | 96 ms maximum | 120 mm offset | Below 200 ms deadline |
| Output voltage at 80 mm | \(359.6\pm0.8\) V | 3.3 kW | Regulation maintained |
| Core temperature at 80 mm | 45.2 °C after 20 min | Tracking enabled | Improved thermal margin |
| Relay-copper loss share | 31.2% | 80 mm, three-coil | Dominant loss bottleneck |
| Third harmonic | 7.6% before mitigation | 80 mm, 3.3 kW | Requires harmonic control |
| Third harmonic after notch | 2.4% | 80 mm, 3.3 kW | 0.6 percentage point gain |
| Duty-cycle step limit | 0.005 | Each update | Limits transient stress |
I interpret the results in three steps. First, the impedance model and the duty-cycle mapping explain why the efficiency gain grows with offset. At perfect alignment, the fixed duty cycle is already close to the efficient load. The controller has little room to improve, and the measured gain is smaller than the expanded uncertainty. As the offset increases, the fixed duty cycle becomes increasingly mismatched. The reflected resistance falls, the primary current rises, and the loss distribution shifts. The load-matching loop recovers part of the efficiency by moving the reflected resistance back toward the efficient interval. This is why the gain is small at 0 mm, moderate at 40 mm, and large at 80 mm and 120 mm.
Second, the identification results show that phase-based estimation with Kalman filtering is adequate for an electric vehicle wireless charging controller. The filter reduces the error standard deviation by about 60% at all tested signal-to-noise ratios. The residual test confirms that the covariance settings are not arbitrary. The convergence time is below 20 ms, which is much shorter than the outer control period and much shorter than the parking maneuver time scale. The zone hysteresis prevents the controller from switching unnecessarily when the estimate approaches a threshold. This is important because a duty-cycle step produces a current transient, and repeated steps would increase thermal and electrical stress.
Third, the three-coil results show that control optimization has a limit. The relay coil can improve the magnetic field, but its copper loss becomes the largest single loss component at 80 mm offset. The third harmonic is not negligible. A notch branch and a better winding reduce the loss, but they do not eliminate the relay-coil loss. For an electric vehicle wireless charging system, the control strategy should therefore be coordinated with the coil design. If the relay-coil loss is dominant, a higher control gain will not produce a proportional efficiency gain. The controller should instead reduce the output in the weak-coupling zone and request mechanical alignment. This is consistent with the boundary limits I impose.
I also consider the practical implications for an electric vehicle charging interface. The controller needs a reliable estimate of mutual inductance, but it also needs a safe response when the estimate is uncertain. I use three mechanisms to handle uncertainty. The first is the Kalman filter, which reduces the effect of measurement noise. The second is the zone hysteresis, which prevents threshold chattering. The third is the power-limit mode, which protects the converter when the coupling is too weak or the temperature is too high. These mechanisms are complementary. The filter improves the estimate, the hysteresis stabilizes the decision, and the power limit protects the hardware. I do not rely on a single mechanism to guarantee safe operation.
I further examine the effect of the duty-cycle slew-rate limit. A limit of 0.005 per update appears conservative, but it has two benefits. It keeps the secondary current transient within the safe operating area of the rectifier, and it prevents the load-matching loop from interacting with the resonant network. The inner loop updates every 100 μs, while the outer loop updates every 5 ms. The separation of time scales is deliberate. The inner loop regulates the output voltage, and the outer loop adjusts the load match. If the outer loop were too fast, it would fight the inner loop and produce limit cycles. If it were too slow, the system would remain at a mismatched load during a parking maneuver. The 5 ms outer period is a compromise that I validate through the response-time measurements.
I note that the efficiency gain at 0 mm is not statistically meaningful under the stated uncertainty. This is not a weakness of the method; it is a correct interpretation of the measurement. The method is designed for offset conditions, and the 0 mm case is already close to the fixed-parameter optimum. Reporting a small gain as a success would overstate the result. The 40 mm, 80 mm, and 120 mm cases have gains that exceed the uncertainty boundary, and the 80 mm case is the most relevant for an electric vehicle wireless charging system because it represents a typical lateral parking error. The output voltage remains within regulation, and the core temperature decreases after 20 minutes of operation.
I also compare the two-coil and three-coil cases qualitatively. In the two-coil case, the dominant loss at offset is the primary and secondary copper loss, and the load-matching loop can recover a significant part of the efficiency. In the three-coil case, the relay-copper loss is dominant at 80 mm offset, and the efficiency gain is constrained by that loss. The control strategy is the same, but the achievable gain is different. This is why I report the loss decomposition together with the efficiency. A single efficiency number does not show where the remaining loss is located, and it does not show whether a further control improvement is possible.
I include the harmonic loss because it is coupled to the coil design and the control. The third harmonic is 7.6% of the fundamental before mitigation, and the additional AC loss is about 19 W. A 255 kHz notch branch reduces the third harmonic to 2.4%, reduces the relay copper loss by about 24 W, and improves the efficiency by 0.6 percentage points. This improvement is smaller than the 4.3 percentage point gain from load matching at 80 mm, but it is still relevant because it reduces the thermal stress in the relay coil. In an electric vehicle wireless charging system, thermal margin is a design constraint, not a secondary metric. A lower relay temperature can allow a higher continuous power rating or a longer component life.
I summarize the control law in a compact form. The outer loop estimates \(M\) and selects a zone. The middle loop computes a target duty cycle \(D^*\) from the efficiency map. The inner loop regulates the output voltage. The duty-cycle command is rate-limited:
$$D_{k}=D_{k-1}+\mathrm{sat}\left(D^* – D_{k-1},\;-\Delta D_{max},\;\Delta D_{max}\right),$$
where \(\Delta D_{max}=0.005\). The zone decision is
$$Z_k=\begin{cases}
\text{normal}, & \hat{M}_k\geq 26\,\mu\mathrm{H},\\
\text{mild}, & 22\leq\hat{M}_k<26\,\mu\mathrm{H},\\
\text{moderate}, & 17\leq\hat{M}_k<22\,\mu\mathrm{H},\\
\text{weak}, & \hat{M}_k<17\,\mu\mathrm{H}.
\end{cases}$$
The zone transition is allowed only when the last three estimates agree. I write the transition condition as
$$\left|Z_k-Z_{k-1}\right|>0 \quad \text{and} \quad \sum_{i=0}^{2}\mathbf{1}\left(Z_{k-i}=Z_k\right)=3.$$
This condition is simple to implement and easy to audit. It also prevents a single noisy estimate from changing the control policy. I prefer this deterministic hysteresis to a soft blending rule because the efficiency map has distinct regions, and a soft blend would require a continuous loss model that is difficult to validate at every operating point.
I also express the power-limit condition. The controller enters power-limit mode when any of the following is true:
$$I_1>1.1I_{1,rated},$$
$$T_{core}\geq 60\,^{\circ}\mathrm{C},$$
$$T_{relay}\geq 65\,^{\circ}\mathrm{C}.$$
In power-limit mode, the output command is reduced, and the duty-cycle step limit is tightened. The controller exits power-limit mode only after the current and temperatures fall below their release thresholds for a defined hold time. This prevents repeated entry and exit around a single threshold. The release thresholds are lower than the entry thresholds, which provides hysteresis in the thermal and current protection loops.
I consider the limitations of the study. The mutual-inductance identification is validated in simulation and with repeated power measurements, but the phase extraction is sensitive to sensor alignment and to the compensation tolerance. I remove the calibrated reactive component before computing the estimate, but a large compensation drift could still bias the result. The three-coil loss decomposition is based on a specific coil geometry and a specific relay-coil position. A different geometry could change the loss shares. The efficiency map is offline and is interpolated online. If the battery resistance changes significantly with temperature or state of charge, the map should be updated. These limitations do not invalidate the control strategy, but they define the conditions under which the reported gains are expected to hold.
I propose three extensions for future work. First, I would include vehicle localization information in the zone decision. If the electric vehicle position is known from a camera or an ultrasonic sensor, the controller can anticipate a weak-coupling condition before the mutual-inductance estimate crosses the threshold. This would reduce the response time and improve the alignment request. Second, I would include battery temperature in the load-matching map. The battery resistance changes with temperature, and the optimal duty cycle depends on the battery resistance. A temperature-aware map would improve the efficiency at low or high ambient temperatures. Third, I would coordinate the relay-coil design with the harmonic-mitigation branch. The relay-copper loss is the dominant loss at 80 mm offset, so a coil design that reduces the relay current or improves the current distribution could produce a larger gain than a further refinement of the control loop.
I conclude that the efficiency tracking problem for an electric vehicle wireless charging system is a coupled estimation-and-matching problem. The mutual inductance determines the reflected impedance, the reflected impedance determines the efficient load interval, and the DC/DC duty cycle determines the equivalent AC load. I use a sliding discrete Fourier transform to extract the fundamental phase, a Kalman filter to estimate mutual inductance, a four-zone hysteresis logic to stabilize the coupling decision, and a hierarchical controller to match the load. The simulation and measurement results show that the efficiency at 80 mm offset increases from 86.5% to 90.8%, the mutual-inductance identification error standard deviation at 20 dB SNR is reduced by about 61%, and the response time remains below 96 ms. The three-coil system shows that the efficiency gain is constrained by the relay-copper loss and harmonic loss. The control strategy is therefore necessary but not sufficient: it must be combined with a coil design that manages the relay-coil loss and a harmonic-mitigation network that reduces the additional AC loss. For an electric vehicle wireless charging interface, this combination is the practical path to a robust and efficient charging process under realistic parking misalignment.