Wireless Power Transfer

Introduction

Fig. 1: Schematics for inductive WPT. (Image source: Y. Liu.)

Wireless Power Transfer (WPT) includes diverse approaches such as inductive, capacitive, laser, radio-frequency, and acoustic methods. [1] This report focuses specifically on inductive WPT (see Fig. 1) in comparison to radio-frequency alternatives. Given the widespread adoption of inductive charging in modern portable devices like smartphones and smartwatches, we will examine the key advantages.

Comparison

Inductive WPT is distinguished by its high transmission efficiency and robust tolerance for physical obstructions. Experimental results have demonstrated efficiencies exceeding 90% at a distance of 75 cm. [2] In contrast, alternative systems such as radio-frequency (RF) WPT have achieved only 19% efficiency at a much shorter range of 1 cm. [3] Furthermore, researchers observed negligible efficiency degradation when human bodies or everyday objects intercepted the transmission path. [2] Unlike laser WPT, which strictly requires an uninterrupted line-of-sight to maintain power transfer, inductive systems remain effective despite barriers. These attributes position inductive WPT as a promising candidate for long-range applications, including room-scale charging.

Radio-Frequency WPT

A natural question arises regarding the superior efficiency of inductive WPT compared to other methods. This efficiency stems from the nature of magnetic coupling. The magnetic field is highly confined, and power is transferred directly to the receiving coil. In contrast, RF systems typically rely on radiative propagation, where energy dissipates as it spreads over a distance. This rapid decay in power density for RF systems is described by the equation [4]

Here P_r, P_t, A_r, A_t, λ and d represent the received power, the transmitted power, effective area of the receiving antenna, effective area of the transmitting antenna, the wavelength and the distance between antennas. The left-hand side of the equation represents the transmission efficiency, accounting solely for propagation loss.

We notice in this case that the efficiency decreases inversely with the square of the distance between the two antennas. This implies that an efficiency of 19% at 1 cm would drop to approximately 0.0019% at 1 m. Consequently, to deliver a standard 5 W charge to a smartphone at this efficiency, the transmitting antenna would require a 263 kW power supply. Such a requirement is not only impractical but also poses significant risks of electromagnetic interference with surrounding objects.

Inductive WPT

In contrast to radiative systems, inductive WPT uses coupled antennas to transmit energy. This allows for higher efficiency by minimizing radiative losses. However, the system's efficiency is limited by signal reflections caused by impedance mismatches. To address this, we must first understand the concept of impedance. In DC circuits, resistance simply opposes the flow of current. In AC circuits, however, components not only reduce voltage but also introduce a phase shift to the wave. Therefore, we define impedance Z as a complex number that encapsulates both the magnitude change and the phase difference

where V and I represent voltage and current respectively, i is the imaginary unit, and θ is the phase difference between the voltage and current. The impedance for standard circuit components is given by

where ω is the angular frequency of the AC signal, and R, C, and L denote resistance, capacitance, and inductance, respectively. We observe that a resistor is purely real and therefore does not alter the phase of the signal. Consequently, controlling the phase requires the use of inductors and capacitors. For example, connecting an inductor and a capacitor in parallel creates a tank circuit, the total impedance of which is

By varying the capacitance C or inductance L, we can tune the impedance of this tank circuit from negative pure imaginary to positive pure imaginary at given ω. This tunability is critical for optimizing power transfer efficiency. In WPT systems, power is delivered from a source to the antennas via transmission lines. Because these electrical signals propagate as waves, they behave similarly to sound waves in a tunnel: if the wave encounters the terminal (analogous to a wall), a portion of the energy reflects back toward the source rather than being completely absorbed. The reflection coefficient Γ quantifies the fraction of voltage reflected:

Here, Z₀ represents the impedance of the transmission line, known as the square root of the ratio of the line's inductance per unit length to its capacitance per unit length (purely real). Z_L denotes the impedance of the connected system. From the reflection coefficient equation, it is evident that when Z_L equals Z₀ (impedance matching), the reflection coefficient vanishes. This condition ensures maximum power transfer from the source to the load. To realize this optimal state, we must first know the exact expression for the system impedance, Z_L.

In an inductive WPT configuration, the impedance Z_L seen by the transmission lines is given by

Here, Z₁ and Z₂ represent the impedances of the tank circuits at the transmitting and receiving sides, respectively. L₁ and L₂ denote the inductances of the transmitting and receiving antennas, while L₁₂ is the mutual inductance between them. The system operates at an angular frequency ω, with R₂ representing the resistive load at the receiver.

The equation for the system's total input impedance consists of two parts. The first term, (Z₁ + iωL₁), accounts for the impedance of the transmitter antennas and its local tank circuit. The second term represents the impedance originating from the receiver. This contribution depends on the mutual inductance L₁₂, which quantifies the strength of the magnetic coupling between the two antennas. Summing these components yields the total input impedance seen by the source.

To verify that matching the load impedance Z_L to the source impedance Z₀ is possible, we consider a simplified case where the receiver load is purely resistive (R₂). By carefully tuning Z₁ and Z₂, we can cancel the inductance of the antennas. This renders the total system impedance purely real. Consequently, at a specific operating frequency, Z_L can be tuned to exactly equal Z₀, thereby eliminating reflections. Under this optimized condition, inductive WPT becomes significantly more efficient than RF WPT.

Conclusions

We have discussed the advantages of inductive Wireless Power Transfer (WPT) over other wireless power methods. It offers a theoretical explanation for why RF WPT, which also relies on electromagnetic waves, demonstrates significantly lower efficiency due to its radiative nature. Furthermore, it analyzes the impedance matching conditions required to achieve zero-reflection transmission in inductive WPT systems.

© Yidi Liu. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] J. Van Mulders et al. "Wireless Power Transfer: Systems, Circuits, Standards, and Use Cases," Sensors 22, 5573 (2022).

[2] A. Kurs et al., "Wireless Power Transfer via Strongly Coupled Magnetic Resonances," Science 317, 83 (2007).

[3] T. Nusrat et al., "Far-Field Wireless Power Transfer for the Internet of Things," Electronics 12, 207 (2023).

[4] H. T. Friis, "A Note on a Simple Transmission Formula," Proc. IRE 34, 254 (1946).