Elimination of Power Dead Zones in Wireless Power Transfer Systems Using a Hybrid Dual-Receiver Architecture and Adaptive Control

To theoretically quantify the power dead zones identified previously, this section analyzes the magnetic coupling and system modeling.

2.1 Geometrical modeling of the Wireless Power Transfer system

The fundamental limitation of conventional WPT systems is that a single receiver coil cannot deliver sufficient power over a wide range of transmission distances. Large receiver coils experience strong magnetic coupling at short distances, which induces the frequency splitting phenomenon and severely degrades delivered power. In contrast, small receiver coils exhibit weak mutual coupling at longer distances, resulting in insufficient received power. This inherent trade-off motivates the use of a hybrid receiver architecture capable of maintaining stable power delivery across varying coupling conditions.

Figure 1 shows the Ansys Maxwell 3D model of the proposed transmitter and nested dual-receiver architecture. The physical configuration consists of a large primary receiver coil and a smaller auxiliary receiver coil concentrically embedded within it, enabling complementary coupling characteristics over different transmission distances.

1.png

Figure 1. ANSYS Maxwell 3D model of the transmitter and the nested dual-receiver architecture

The geometrical and electromagnetic parameters of the transmitter and nested receiver coils are summarized in Table 1, including the inner and outer diameters, number of turns, and calculated self-inductance values.

Table 1. Geometrical and electromagnetic parameters of the designed transmitter and nested receiver coils

The parameters, presented in Table 1, quantitatively define the physical structure of the proposed system and form the basis for establishing its equivalent electrical model. Due to the axial symmetry of the considered coil geometry and the absence of misalignment, a 2D axisymmetric electromagnetic model is adopted for the simulations, while the 3D illustration in Figure 1 is provided solely for visualization purposes.

The magnetic coupling between the transmitter and receiver coils is characterized by the mutual inductance $M$, which depends on the coil dimensions and the transmission distance [26]:

$M(d)=\frac{\mu N_T N_R r_T^2 r_R^2}{\left(r_T^2+r_R^2+d^2\right)^{3 / 2}}$       (1)

Here, $\mu_0$ is the permeability of free space, $N_T$ and $N_R$ denote the number of turns of the transmitter and receiver coils, respectively, $r_T$ and $r_R$ are the coil radii, and $d$ represents the transmission distance. This expression highlights the strong dependence of the coupling strength on both coil radius and separation distance. In particular, larger receiver coils exhibit a rapid increase in mutual inductance as the distance decreases, which plays a critical role in the emergence of over-coupling effects under fixed-frequency operation.

2.2 Received power equation

Based on the parameters listed in Table 1, the equivalent electrical circuit of the proposed hybrid WPT system is established, as shown in Figure 2. A series-series (SS) compensation topology is adopted, where the compensation capacitors $\left(C_{t x}, C_{r x 1}, C_{r x 2}\right)$ are connected in series with the transmitter and receiver coils to cancel reactive omponents at the resonant frequency $f_0$, thereby ensuring maximum active power transfer to the load.

2.png

Figure 2. Equivalent resonant circuit of the proposed system

The steady-state electromagnetic interaction and power flow within the resonant circuit are mathematically described by the simplified impedance matrix shown in Eq. (2). This formulation provides a systematic framework for analyzing the electrical behavior of the transmitter-receiver system under different magnetic coupling conditions.

$\left[\begin{array}{c}V_s \\ 0\end{array}\right]=\left[\begin{array}{cc}Z_s & j w M_{1 i} \\ j w M_{1 i} & Z_i\end{array}\right]\left[\begin{array}{c}I_s \\ I_i\end{array}\right]$       (2)

In Eq. (2), $V_s$ denotes the source voltage amplitude. The variable $I_s$ explicitly represents the primary source current (transmitter current), while $I_i$ represents the induced secondary current flowing in the active receiver coil (i = 1 for the large coil, and i = 2 for the small coil). The terms $Z_S$ and $Z i$ correspond to the self-impedances of the transmitter and receiver, and $M_{1 i}$ represents the mutual inductance between them.

This formulation allows the system behavior to be analyzed using the concept of reflected impedance, Z_ref,i, which represents the impedance of the receiver as seen from the transmitter side. The reflected impedance is the key physical parameter governing the frequency splitting phenomenon in strongly coupled resonant WPT systems [27, 28].

$Z_{\text {ref}, i}=\frac{\left(w M_{1, i}\right)^2}{Z_i}$       (3)

where, $\omega=2 \pi f_0$ defines the angular operating frequency. By substituting the reflected impedance $Z_{\text {ref}, i}$ into Kirchhoff's voltage law for the primary circuit, the total input impedance limits the primary source current $I_s$. Consequently, the output power $P_{\text {out}, i}$ delivered to the load is expressed as:

$P_{\text {out }, i}=\underbrace{\left|\frac{V_S}{Z_s+Z_{\text {ref }}}\right|^2}_{\text {square Source current }} \operatorname{Re}\left\{Z_{\text {ref }, i}\right\} \frac{R_L}{R_L+R_i}$        (4)

where, $R_L$ represents the equivalent load resistance, and $R_i$ is the internal parasitic resistance of the active receiver coil ($R_1$ or $R_2$). At the resonance frequency, Eq. (4) becomes:

$P_{\text {out}, i}=\left(\frac{V_S}{R_S+\frac{\left(w M_{1 i}\right)}{R_i+R_L}}\right)^2 \frac{\left(w M_{1 i}\right) R_L}{\left(R_i+R_L\right)^2}$       (5)

Here, $R_s$ denotes the equivalent source resistance accounting for the inherent conduction and parasitic losses of the transmitter-side power source and its driving circuitry.

2.2.1 Dead zone identification

From the received power expression, it can be seen that the reflected impedance increases proportionally to the square of the mutual inductance.

At the resonant frequency $\omega_0$, the secondary impedance becomes purely resistive, and the reflected impedance $Z_{\text {ref}, i}$ is entirely real. This real part, denoted as the reflected resistance $R_{r e f, i}=\operatorname{Re}\left\{Z_{r e f, i}\right\}$, is given by:

$R_{\text {ref}, i}=\left(\omega_0 M_{1 i}\right)^2 /\left(R_L+R_i\right)$       (6)

According to the maximum power transfer theorem, the delivered power peaks at a critical mutual inductance $M_c$, which occurs when the reflected resistance exactly matches the primary equivalent source resistance ($R_{\text {ref}, i}=R_s$). This allows us to analytically define the critical coupling condition:

$M_c=1 / \omega_0 \sqrt{\left(R_L+R_i\right) R_S}$       (7)

The power dead zone emerges when the transmission distance decreases further, pushing the system into the overcoupled regime where $M_{1 i}>M_c$. Under this specific condition, $R_{\text {ref }, i}$ grows quadratically with $M_{1 i}$ and heavily dominates the total input impedance $R_{\text {ref}, i} \gg R_s$. Consequently, the primary source current is severely choked $\left(I_s \approx \frac{V_s}{R_{\text {ref}, i}} \propto 1 / M_{1 . i}^2\right)$, causing the output power to plummet as $P_{\text {out}, i} \propto 1 / M_{1 i}^2$. This mathematical derivation proves that the dead zone is not a vague spatial phenomenon, but a direct consequence of a severe impedance mismatch driven by excessive mutual coupling under a strict fixed-frequency constraint.

Table 2. Simulation parameters and design specifications

To evaluate the proposed hybrid system, the operational characteristics are analyzed at a unified resonance frequency of $f_0$ = 100 kHz. This design choice is fundamental to eliminating reactive power components, allowing the system to be modeled as a purely resistive network at the switching instances. The physical and electrical parameters used in the Maxwell and Python-based simulations are detailed in Table 2. The compensation capacitors required to achieve resonance for each coil are calculated using the following resonance condition:

$C_{R x i}=\frac{1}{\left(2 \pi f_0\right)^2 L_i}$      (8)

where, $L_i$ represents the self-inductance of the transmitter or receiver.

The system parameters are selected based on practical engineering trade-offs. The 100 kHz operating frequency optimally balances coil size and efficiency; lower frequencies would require bulky coils, whereas higher frequencies increase inverter switching losses and parasitic AC losses. Furthermore, the switching point is dynamically determined by the physical intersection of the power profiles of the two receivers, marking the optimal transition boundary between the over-coupled and under-coupled regime.

The analysis begins with the magnetic behaviour shown in Figure 3, which illustrates the variation of mutual inductance with the transmission distance.

3.png

Figure 3. Mutual inductance versus transmission distance

The large receiver (blue curve) exhibits an exponential increase in coupling as the receiver approaches the transmitter. In the deep near-distance region $(d=2 \mathrm{~cm})$, $M_{12}$ rises to approximately 35 μH. This excessive value is the primary cause of system instability, a phenomenon that is significantly exacerbated when the transmitter and receiver coils are symmetrical (identical in size). Conversely, the small receiver (red curve) maintains a stable and moderate profile, with $M_{13}$ remaining below 10 μH even at the closest proximity. This demonstrates that decreasing the receiver's radius effectively mitigates the over-coupling problem by limiting the mutual inductance profile. The Maxwell simulations, along with the fixed electrical parameters summarized in Table 2, collectively determine the system's power delivery performance. This interaction is fundamental in explaining the power collapse observed in Figure 4, where the excessively high mutual inductance of the large receiver at close distances triggers an impedance surge that exceeds the source driving capability, leading to the dead zone phenomenon, as presented in previous work.

4.png

Figure 4. Output power $P_{\text {out}}$ as a function of frequency for (a) the large receiver and (b) the small receiver at different transmission distances

The impact of these coupling profiles on power transfer stability is evident in the frequency response curves in Figure 4(a) and (b). Frequency bifurcation at 2 cm severely attenuates the response at 100 kHz, preventing power transfer. At a close range of 2 cm, the frequency response of the large coil splits into two sidebands, leaving a deep valley exactly at the 100 kHz operating point. This confirms that the large coil is incapable of transferring power at the designed frequency in this region as shown in Figure 4(a). In contrast, the small coil exhibits a robust single-peak response centered at 100 kHz across all distances as shown in Figure 4(b).

The observed performance discrepancy is directly explained by the mathematical derivation presented in Eqs. (4) and (5). At close distances, the mutual inductance of the large receiver, $M_{12}$, becomes excessively high, causing $Z_{r e f, 2}$ to increase drastically. Since $Z_{\text {ref}}$ appears in the squared denominator $\left|Z_S+Z_{\text {ref}, i}\right|^2$, this impedance surge chokes the primary current Is, causing the output power to collapse (the dead zone). The mutual inductance of the small receiver, $M_{13}$ remains lower.

Consequently, $Z_{r e f, 2}$ stays within the optimal range (impedance matching region), maintaining a sufficient source current and ensuring stable power transfer.

Figure 5 highlights the necessity of the proposed hybrid receiver architecture. The standalone large receiver (red curve) shows a pronounced power dead zone at short transmission distances (d < 6 cm) due to the frequency splitting phenomenon. In contrast, the standalone small receiver (blue curve) effectively covers this near-distance region but delivers from reduced power delivery capability at longer distances because of weak magnetic coupling.

The intersection point at approximately d = 9 cm demonstrates a clear complementary relationship between the two receivers. By switching between the large and small receivers based on the coupling condition, the proposed hybrid system maintains a continuous and high output power profile across the entire charging range, effectively eliminating the power dead zone.

5.png

Figure 5. Output power vs. distance highlighting the optimal switching point

To evaluate the effectiveness of the proposed coil-selection algorithm, the system’s transient and steady-state responses were analyzed at the critical switching distance of 9 cm. As shown in Figure 7, the operation is divided into three distinct phases:

System identification phase (0 to 4 ms): During this interval, the algorithm rapidly scans the available receivers. From 0 to 2 ms , small receiver $\left(R_{x 2}\right)$ is activated, followed by the large receiver $\left(R_{x 1}\right)$ from 2 to 4 ms. The observed fluctuations in the output voltage ($V_{\text {out}}$) and current ($I_{\text {out}}$) amplitudes during this phase are result from the diagnostic process used to evaluate the mutual coupling for each coil. The brief inductive spikes at the switching boundaries are inherent to the high rate of change in current $(d i / d t)$ during transitions in inductive power transfer systems. The proposed control strategy introduces a 4 ms diagnostic delay prior to the final receiver selection. This brief sampling window is intentionally incorporated to allow initial electrical transients to settle, ensuring the decision relies strictly on steady-state power values. In practical WPT applications, the mechanical displacement of the receiver is significantly slower than this millisecond-scale electrical evaluation. Consequently, the sensing delay has a negligible impact on the magnetic coupling variations and does not compromise the continuous power delivery or system stability.

7.png

Figure 7. Transient voltage and current at switching point

Decision point and seamless transition (at 4 ms): The most critical aspect of the proposed control logic is its stability at the decision-making point. The micro-zoom insets in Figure 7 highlight the transition at 4 ms. The transition to the optimal receiver is achieved seamlessly, with no significant phase distortion or prolonged oscillation.

Steady State Stability (4-20 ms): after the switching instance, the system maintains a quantitatively stable sinusoidal output regulated exactly at a 20 V peak. Specifically, as evidenced by the micro-zoom insets, the output reaches its steady-state with a rapid settling time of less than 0.5 ms and exhibits a negligible voltage amplitude ripple of under 1%. The uniform waveforms confirm successful locking onto the optimal state with zero observed chattering events during the transition, ensuring high-quality power delivery even at the boundary distance. As observed in Figure 7, the output voltage stabilizes with a peak amplitude $V_{\text {peak}}$ of 20 V. This corresponds to an RMS voltage ($V_{r m s}$) of approximately 14.14 V, which, given the 10 $\Omega$ load, results in a steady-state output power of 20 W. This quantitative result aligns with the analytical power-distance curves presented in Figure 8, where the intersection point at d ≈ 9 cm yields the same power level.

This agreement between the time-domain waveforms and the frequency-domain analysis confirms the effectiveness of the proposed switching logic at the critical operating point. To further validate robustness beyond this boundary condition, the system response was also evaluated at two extreme transmission distances (2 cm and 11 cm), representing over-coupled and under-coupled operating conditions.

Figure 8 shows the transient voltage and current responses of the proposed system at transmission distances of 2 cm and 11 cm, respectively, representing the extreme over-coupled and under-coupled operating conditions. In both cases, the controller initially performs a time-multiplexed evaluation of the available receiver coils by sequentially activating each path and storing the corresponding time-averaged output power values.

8a.png

(a)

8b.png

(b)

Figure 8. Transient voltage and current responses at transmission distances of (a) 2 cm and (b) 11 cm

At a distance of 2 cm, the large receiver experiences severe over-coupling, resulting in significant power reduction, while the small receiver provides a higher and more stable output. After comparing the stored power values, the algorithm directly selects the small receiver ($R_{x 2}$) and latches to this configuration for the remainder of the operation.

Conversely, at a distance of 11 cm, the magnetic coupling is weak, and the small receiver delivers insufficient power. In this case, the stored power comparison prompts the controller to immediately select the large receiver ($R_{x 1}$) as the optimal power transfer path. As shown in both figures, once the decision is made, the selected receiver remains active without further switching, and the voltage and current waveforms converge smoothly to a stable sinusoidal steady state.

This consistent behavior at both extreme distances confirms that the proposed control strategy bases its decisions on stored average power comparisons rather than instantaneous measurements, thus avoiding chattering and ensuring quantitatively robust operation across a wide range of coupling conditions. In this context, robustness is mathematically verified by the system's ability to maintain a guaranteed minimum power delivery threshold of 20 W across the entire 2 cm to 16 cm transmission range, while strictly maintaining a 0% frequency deviation from the 100 kHz resonance point.

Figure 9 shows the output power versus transmission distance for the standalone large receiver, standalone small receiver, and the proposed hybrid system under fixed-frequency operation.

Large receiver limitations: At close distances (d < 5 cm), the large receiver shows a sharp decline in power, nearly reaching zero. This is known as the power dead zone, caused by the frequency splitting phenomenon.

Small receiver performance: The small receiver provides a stable and high power output in the near-field region (0 to 7 cm), effectively filling the gap left by the large receiver. However, its power decays rapidly beyond 9 cm due to weak coupling.

Proposed hybrid system: The orange curve represents the hybrid system controlled by Python. By dynamically switching at the optimal crossover point (approximately 9 cm), the system maintains a continuous and high-power level throughout the entire range, successfully eliminating the dead zone. The proposed hybrid system establishes a continuous power path, ensuring that the delivered power never drops below a critical threshold (approximately 20 W) even at zero distance. This eliminates the power discontinuities typical of single receiver systems, providing a reliable charging profile regardless of transmitter-receiver proximity. Figure 10 illustrates the 3D power profile of the system under the proposed control strategy, mapping the output power (z-axis) as a simultaneous function of operating frequency (x-axis) and transmission distance (y-axis). The color gradient, mapped from dark blue (indicating near-zero power in the dead zones) to yellow (indicating peak power delivery up to 35 W), visually quantifies the energy transfer capability. A defining characteristic of this surface topology is the continuous, unbroken ridge (the yellow crest) that aligns strictly along the fundamental resonant frequency ($f_0$ = 100 kHz) across the entire transmission range.

9.png

Figure 9. Output power vs. distance for hybrid and standalone systems

10.png

Figure 10. Output power versus distance for standalone and hybrid receiver configurations

Unlike conventional strongly coupled systems where the resonant peak bifurcates, the proposed hybrid switching mechanism effectively suppresses this splitting phenomenon. Consequently, the system maintains a unimodal, high-power response regardless of variations in transmission distance. As shown by this surface, the algorithm renders the system architecturally immune to frequency splitting, thereby eliminating the need for complex and costly frequency-tracking or active impedance-matching circuitry. The locking of the peak power at exactly 100 kHz validates that the control algorithm successfully enforces resonant operation, ensuring a stable and efficient energy link throughout the charging process.

Figure 11 highlights the inherent trade-off between efficiency and power continuity in fixed-frequency WPT systems.

11.png

Figure 11. Efficiency versus distance for standalone and hybrid Wireless Power Transfer (WPT) systems

While the proposed hybrid system successfully eliminates the power dead zone, a moderate reduction in overall efficiency is observed when operating with the small receiver compared to the theoretical peak efficiency of the large receiver at optimal distances. This behavior is an intentional design choice, as the large receiver would otherwise deliver near-zero power in the over-coupled region due to frequency splitting. By prioritizing power continuity and fixed-frequency operation over peak theoretical efficiency, the delivery across the entire charging range, which is of greater practical importance for real-world charging applications.

Table 3 summarizes the key performance differences between the standalone receiver configurations and the proposed hybrid system under fixed-frequency operation. The comparison shows that while each standalone receiver is effective over a limited distance range, only the hybrid architecture ensures continuous power delivery across the entire charging range without requiring frequency tuning or active impedance matching.

To transition the proposed hybrid dual-receiver architecture from the co-simulation environment to a physical prototype, several critical engineering considerations must be addressed. Operating strictly at 100 kHz requires the use of wide-bandgap semiconductors, such as Silicon Carbide (SiC) or Gallium Nitride (GaN) MOSFETs, to minimize switching losses and thermal dissipation in the primary inverter. Furthermore, the real-time power evaluation algorithm necessitates a microcontroller equipped with a high-speed Analog-to-Digital Converter (ADC). The sampling window must be tightly optimized to ensure the computation delay remains well below the AC wave period (10 ms), thereby preventing any transient lag during the critical switching event at the 9 cm boundary. Finally, the physical layout of the concentrically nested coils must account for high-frequency skin and proximity effects by utilizing individually insulated Litz wire, ensuring that the intrinsic quality factor remains high for both the primary and auxiliary receivers.

Table 3. Comparative performance evaluation of Large receiver, Small Receiver, and proposed hybrid system