**3. Efficiency and characteristic impedance variation simulations with different software tools**

Resonance frequency is a key parameter in system design, and the value can be changed by adjusting the distance between the transmission and characteristic impedance of the electrical circuit. First, the electrical circuit is created in a PSIM circuit simulator and the resulting voltage and current waveforms are obtained via the simulator. PSIM is simulation software designed for power electronics, motor control, and dynamic system simulation. The simulation is solved in the Matlab platform, and a procedure to calculate the parameters of the equivalent circuit is performed in Maxwell. For an air gap of 10 cm L = 999.2nH, C = 124pF, Lm = 128.6H, Z0 = 5Ω, the efficiency chart and variations in the equivalent impedance chart are given in **Figures 5** and **6**, respectively.

**Figure 5.** Efficiency chart.

Since voltage and current are electrical quantities, the voltage equation can be written in a manner that calculates the electrical efficiency [18]. This leads to a set of equivalent impedance equations. The equivalent impedance is obtained by (19). Assuming C = C1 = C2 in the resonance

> *out out out in in Eq P IZ P IZ*

2

<sup>2</sup> (20)

1 (21)

*m*

w

(22)

*j C*

w

2 0

1

w

æ ö ç ÷

ç ÷ æ ö è ø <sup>ç</sup> + ++ ç ÷ ç ÷ <sup>÷</sup> è è è ø øø

h= =

*out m*

w

*jL Z*

**( )**<sup>2</sup> <sup>0</sup>

<sup>=</sup> æ ö

*<sup>I</sup> jL Z R j C*

w

w

+ ++ ç ÷ è ø 2 0

*j C <sup>L</sup> R jL j C jL Z R*

++ + ç ÷

At a given resonant frequency, the conditions for system efficiency are defined for three states,

w

1

w

2 0 2 2 1

<sup>=</sup> æ ö æ ö + ++ ç ÷ ç ÷ è ø æ ö

*I jL*

*in*

Eqs. (21) and (14) are substituted for Eq. (20);

*m*

w

*jL Z R*

w

defined by Eqs. (23), (24), and (25).

1

w

coupling system, the efficiency can be defined by (22).

56 Wireless Power Transfer - Fundamentals and Technologies

**Figure 4.** Magnetic coupling circuit.

**2.2. Efficiency equation**

Eq. (11) makes use of (21);

h

**Figure 6.** Equivalent impedance chart.

The transmitter and receiver coils have four different resonance coupling states: In the first state, both the transmitter and receiver coil loops are not resonant; in the second, the transmitter coil loop is in resonance, while the receiver coil loop is not; in the third, the transmitter coil loop is not resonant, while the receiver coil is; and in the last state, both the transmitter and receiver coil loops are resonant. When the whole coupling system is in resonance, the impe‐ dance value is at a minimum.

**Figure 7.** Function of efficiency according to mutual inductance and frequency.

The resonant frequencies change from two points to one depending on the length of the air gap. The double-resonance frequency region occurs at low impedance and short range. As the air gap distance and impedance increase, one resonance region appears. At this operation range, the efficiency falls sharply. The critical transmission efficiency would be the same as the peak efficiency over a coupled range.

The distance between the coils responsible for energy transfer should be kept at an optimum value since any increase in the air gap value will degrade the energy transfer ratio. This problem can be solved by optimizing the relation between the frequency and the quality factor.

Calculation of the parameters for the equivalent circuit was carried out in the Maxwell 3D software platform as well as in PSIM.

#### **3.1. Maxwell 3D software simulation results**

Hundred volts were used for the ideal sine source of 13.552 MHz. An inductance of winding value of 999.2 nH, mutual inductance of 128.6 nH, 124 pF capacitor values, R value of 0.22 Ω, and Z0 impedance value of 5 Ω were selected. The circuit scheme and modeled coils can be seen from **Figures 8** and **9**. The magnetic flux density graphs and waveforms of currents and voltage for air gap of 10 cm can be seen from **Figures 10**–**14**.

**Figure 8.** Maxwell 3D circuit scheme.

**Figure 6.** Equivalent impedance chart.

58 Wireless Power Transfer - Fundamentals and Technologies

dance value is at a minimum.

**Figure 7.** Function of efficiency according to mutual inductance and frequency.

the peak efficiency over a coupled range.

The transmitter and receiver coils have four different resonance coupling states: In the first state, both the transmitter and receiver coil loops are not resonant; in the second, the transmitter coil loop is in resonance, while the receiver coil loop is not; in the third, the transmitter coil loop is not resonant, while the receiver coil is; and in the last state, both the transmitter and receiver coil loops are resonant. When the whole coupling system is in resonance, the impe‐

The resonant frequencies change from two points to one depending on the length of the air gap. The double-resonance frequency region occurs at low impedance and short range. As the air gap distance and impedance increase, one resonance region appears. At this operation range, the efficiency falls sharply. The critical transmission efficiency would be the same as

**Figure 9.** Receiver and transmitter coil for 10-cm air gap in Maxwell 3D.

**Figure 10.** Magnetic flux density of receiver and transmitter coil for 10-cm air gap in Maxwell 3D.

**Figure 11.** Maxwell 3D input and output voltages for 10-cm air gap and characteristic impedance of 5 Ω.

**Figure 12.** Maxwell 3D current passing through primary and secondary windings.

**Figure 13.** Maxwell 3D input and output power for 10-cm air gap and characteristic impedance of 5 Ω.

**Figure 14.** Maxwell 3D primary and secondary winding terminal voltage for 10-cm air gap and the characteristic impe‐ dance of 5 Ω.

#### **3.2. PSIM software simulation results**

**Figure 10.** Magnetic flux density of receiver and transmitter coil for 10-cm air gap in Maxwell 3D.

60 Wireless Power Transfer - Fundamentals and Technologies

**Figure 11.** Maxwell 3D input and output voltages for 10-cm air gap and characteristic impedance of 5 Ω.

**Figure 12.** Maxwell 3D current passing through primary and secondary windings.

To determine the power and efficiency, the previously described methodology was fol‐ lowed. First, the mean required values for the circuit parameters for the Maxwell field simu‐ lator were computed. In the circuit simulation, the wireless power transfer system is driven by a sinusoidal voltage source with amplitude of 100 V. **Figure 15** shows the structure of the direct fed wireless power transfer. In order to illustrate how the model works, transient sim‐ ulations were performed with the PSIM circuit simulator.

**Figure 15.** PSIM circuit scheme.

**Figure 16.** Air gap of 10 cm, characteristic impedance of 5 Ω at 13.56 MHz input voltage (VP1 red line) and device voltage waveforms (VP2 blue line).

**Figure 17.** Air gap of 10 cm, characteristic impedance of 5 Ω at 13.56 MHz input current (VP1 red line) and device input waveforms (VP2 blue line).

**Figure 18.** Air gap of 10 cm, characteristic impedance of 5 Ω at 13.56 MHz (blue: input power, red: received power).

**Figure 19.** Magnetic resonance efficiencies according to software platforms.

**Figure 15.** PSIM circuit scheme.

62 Wireless Power Transfer - Fundamentals and Technologies

voltage waveforms (VP2 blue line).

input waveforms (VP2 blue line).

**Figure 16.** Air gap of 10 cm, characteristic impedance of 5 Ω at 13.56 MHz input voltage (VP1 red line) and device

**Figure 17.** Air gap of 10 cm, characteristic impedance of 5 Ω at 13.56 MHz input current (VP1 red line) and device

The transmitting current was 13 A, and the receiving current was 12.88 A for a supply voltage of 70.71 V and the device voltage below 65 V. When input power was 919.2 VA, the amount of power delivery is 837 VA. Approximately 82.2 VA dissipated for losses and the transmitter required 0.098 W (an overhead loss) plus an additional input power of 1.098 VA for every additional 1 VA of power at the receiver.

When the same simulations run on Maxwell and PSIM software platforms for various air gap values, it is observed that for strongly magnetic coupled range up to air gap of 10 cm, efficiency values can be obtained similar. However, if the magnetic coupling gets loosely by the effect of elongated air gap distance, efficiency value differs. The reason of that is the numerical solution method. Therefore, numerical computing such as circuit simulators can calculate the quantities for strongly magnetic resonance couplings.

### **4. Conclusion**

In this study, analysis of voltage and current waveforms in terms of magnetic resonance coupling is performed using numerical computing and a circuit simulator to define the efficiency of wireless power transfer in the time domain. This approach takes into considera‐ tion the nonlinear effects of power losses. The numerical results based on various air gap values are determined from equivalent circuit parameters which are obtained directly from Maxwell by the results of method of moments electromagnetics analysis. The calculation of mutual inductance between two self-resonators is also performed using the Maxwell software, and the equivalent circuit is solved in the circuit simulator PSIM platform.

The aim of this research was to define the efficiency according to the coefficients of variations for the WPT system. The parameters of the system affect the coupling coefficient; Lm is the mutual inductance parameter, while L1 and L2 are nonlinear loss resistance values that depend on the frequency and characteristic impedance of the system. The results were validated using the finite element method.

We concluded that equivalent circuit analysis by means of numerical computing is appropriate for determining the voltage and current waveforms. Additionally, transmission efficiency at a different distance range can be calculated based on the electrical relationship. Efficiency results with respect to load variation show that there are double-resonance frequency regions as well as one-resonance region. The resonant frequencies change from two points to one point depending on the length of the air gap. The double-resonance frequency region occurs at low impedance and short range. As the air-gap distance and impedance increase, one resonance region appears. The efficiency falls sharply at this operation range.

#### **Author details**

Ali Agcal, Selin Ozcira\* and Nur Bekiroglu

\*Address all correspondence to: selinozcira@gmail.com

Department of Electrical Engineering, Yildiz Technical University, Istanbul, Turkey

#### **References**


sion efficiency for wireless power transmission. In: IEICE Tech. Committee Meeting; May 2012; Yokohama. IEICE; 2012. p. 1–5.

[3] Chih-Jung C., Tah-Hsiung C., Chih-Lung L. and Zeui-Chown J. A study of loosely coupled coils for wireless power transfer. IEEE Trans. Circ. Syst. II Express Briefs. 2010;57(7):536–540. doi:10.1109/TCSII.2010.2048403

**4. Conclusion**

64 Wireless Power Transfer - Fundamentals and Technologies

the finite element method.

**Author details**

**References**

Ali Agcal, Selin Ozcira\*

In this study, analysis of voltage and current waveforms in terms of magnetic resonance coupling is performed using numerical computing and a circuit simulator to define the efficiency of wireless power transfer in the time domain. This approach takes into considera‐ tion the nonlinear effects of power losses. The numerical results based on various air gap values are determined from equivalent circuit parameters which are obtained directly from Maxwell by the results of method of moments electromagnetics analysis. The calculation of mutual inductance between two self-resonators is also performed using the Maxwell software, and

The aim of this research was to define the efficiency according to the coefficients of variations for the WPT system. The parameters of the system affect the coupling coefficient; Lm is the mutual inductance parameter, while L1 and L2 are nonlinear loss resistance values that depend on the frequency and characteristic impedance of the system. The results were validated using

We concluded that equivalent circuit analysis by means of numerical computing is appropriate for determining the voltage and current waveforms. Additionally, transmission efficiency at a different distance range can be calculated based on the electrical relationship. Efficiency results with respect to load variation show that there are double-resonance frequency regions as well as one-resonance region. The resonant frequencies change from two points to one point depending on the length of the air gap. The double-resonance frequency region occurs at low impedance and short range. As the air-gap distance and impedance increase, one resonance

the equivalent circuit is solved in the circuit simulator PSIM platform.

region appears. The efficiency falls sharply at this operation range.

and Nur Bekiroglu

Department of Electrical Engineering, Yildiz Technical University, Istanbul, Turkey

[1] Sample A. P., Waters B. H., Wisdom S. T. and Smith J. R. Enabling seamless wireless power delivery in dynamic environments. Proc. IEEE. 2013;101(6):1343–1358. doi:

[2] Wei W., Narusue Y., Kawahara Y., Kobayashi N., Fukuda H. and Tsukagoshi T. Characteristic analysis on double side spiral resonator's thickness effect on transmis‐

\*Address all correspondence to: selinozcira@gmail.com

10.1109/JPROC.2013.2252453


conversion congress and exposition (ECCE); Sep. 2014; Pittsburg, USA: IEEE; 2014. p. 1778–1782. doi:10.1109/ECCE.2014.6953633

