**2. Magnetic resonance circuit**

**Keywords:** wireless power transfer, magnetically coupling resonators, efficiency var‐

Wireless power transfer (WPT) has become popular recently and is expected to be used in plenty of technological devices. The main reason for the recent intense interest in WPT is the sharp increase in the use of electrical devices with various sizes and types of batteries. The main purpose of WPT research was the transfer of as much power as possible, with the goal of high system

In the literature, one of the popular publications that take range variations into consideration is [1]. Critical coupling, frequency splitting, and impedance matching issues were analyzed in [1] using single-turn axillary loops to amplify the magnetic coupling added to resonators. In [1], the authors revealed 50% efficiency for their system, which is suitable for various positions

It has become widely accepted to use a series resonating circuit as the equivalent circuit of a resonator to conduct analysis for resonant frequency [2]. In this context, different attempts have been made to develop a sufficiently precise model [3,4]. There have been many methods employed such as modification circuits and magnetic design of the core properties [5]. In those studies, the scattering parameters were calculated by two port network theory using a network

The state space model was created to obtain an accurate mathematical model of wireless power transfer (WPT) in [6,7], in which the transfer function of the bidirectional IPT system was obtained. As the Ziegler–Nichols method has high overshoot, it was optimized the PID parameters with a phase margin of 60° [8]. The PID parameters were optimized using the genetic algorithm in order to achieve better transient performance in [9]. The dynamic behavior ought to be as fast as possible in [10], in which the H-bridge circuit structure was inserted into the dynamic model. In [11], authors proposed a dynamic model for a multi-pickup system. To overcome the higher order problem and make a straightforward real dynamic model from the energy point of view, it was built a dynamic model of the WPT including a nonlinear inverter

In [13], authors proposed a design methodology using a series-series (S-S) contactless power transfer system for an EV battery charger with two fixed operating frequencies. The converter operates at one of the fixed frequencies for a load-independent current output and at the other operating frequency for a load-independent voltage output. It was proposed hybrid topologies using either SS and PS compensation or SP and PP compensation for battery charging [14]. Controlling output current or voltage is achieved with two different methods. One is PWM (Pulse width modulation) control at a constant frequency at which huge changes in the duty

efficiency in spite of low mutual inductance between coils.

iations

50 Wireless Power Transfer - Fundamentals and Technologies

**1. Introduction**

of the receiver.

analyzer.

and rectifier in [12].

Resonance is observed in many different ways in nature. In general, resonance denotes the oscillation of energy between two different modes. For example, a mechanical pendulum oscillates between potential and kinetic forms of energy. While a system is in resonance, a huge amount of energy can be store with lower excitation. If the energy intake speed ratio of the system is larger than the energy loss ratio of the system, energy accumulation occurs.

**Figure 1.** Resonator.

An example of an electro-magnetic resonator circuit with a coil, a capacitor, and a resistor is shown in **Figure 1**.

In this circuit, the energy oscillates between the coil (which stores energy in the magnetic field) and capacitors (which store energy in an electric field) at a certain resonance frequency.

$$
\alpha\_{\oplus} = \frac{\mathfrak{l}}{\sqrt{LC}} \tag{1}
$$

$$Q = \sqrt{\frac{L}{C}} \frac{1}{R} = \frac{\alpha\_0 L}{R} \tag{2}$$

From Eq. (2), it can be seen that the quality factor of the system increases, decreasing the circuit loss (the reduction of R).

In a high-resonance wireless power transfer system, the resonator system must have a high quality factor for efficient energy transfer. High quality factor electromagnetic resonators are normally made from the conductive components which have relatively narrow resonant frequency widths. The resonance frequency range is narrow, and resonators can be designed to reduce their interactions with foreign objects.

If two resonators are placed close to one another, the resonators can form a link and will be able to exchange energy. The efficiency of the energy exchange varies depending on each resonator and the coupling ratio k. The dynamics of a system with two resonators can be identified from the coupling mode theory or from equivalent circuit analysis of the connection system for the resonator. The equivalent circuit for coupled resonators has a series resonance circuit structure as shown in **Figure 2**.

**Figure 2.** Equivalent circuit of coupled resonator system.

Here, R is the source internal resistance and the frequency is ω(2πf). Vg is called amplitude of the voltage source. Ls and Ld are source and device resonator coils, respectively. And mutual inductance is indicated by *M* =*k L <sup>s</sup>L <sup>d</sup>* . Each coil has a series capacitor that forms a resonator. Rs and Rd resistors denote unwanted resistance (including ohmic and radiation losses) in the coil and the resonance capacitor for each resonator. AC load resistance is shown as RL.

The yield for this circuit can be determined from the ratio of transmitted power for load resistance to the maximum available power of the source, while resonator system is strongly coupled.

#### Wireless Power Transfer by Using Magnetically Coupled Resonators http://dx.doi.org/10.5772/64031 53

$$\frac{P\_L}{P\_{g,max}} = \frac{4U^2 \frac{R\_g}{R\_s} \frac{R\_L}{R\_d}}{\left[ \left( 1 + \frac{R\_g}{R\_s} \right) \left( 1 + \frac{R\_L}{R\_d} \right) + U^2 \right]^2} \tag{3}$$

$$U = \frac{\alpha M}{\sqrt{R\_s R\_d}} = k \sqrt{\mathcal{Q}\_s \mathcal{Q}\_d} \tag{4}$$

To provide the best system performance, proper load and source resistance can be selected or other resistance values can be captured using an impedance matching link. If the resistance is selected using Eq. (5),

$$\frac{R\_g}{R\_s} = \frac{R\_L}{R\_d} = \sqrt{1 + U^2} \tag{5}$$

power transfer efficiency can be expressed by Eq. (6)

*<sup>L</sup> ω L <sup>Q</sup> CR R* = = <sup>1</sup> <sup>0</sup> (2)

From Eq. (2), it can be seen that the quality factor of the system increases, decreasing the circuit

In a high-resonance wireless power transfer system, the resonator system must have a high quality factor for efficient energy transfer. High quality factor electromagnetic resonators are normally made from the conductive components which have relatively narrow resonant frequency widths. The resonance frequency range is narrow, and resonators can be designed

If two resonators are placed close to one another, the resonators can form a link and will be able to exchange energy. The efficiency of the energy exchange varies depending on each resonator and the coupling ratio k. The dynamics of a system with two resonators can be identified from the coupling mode theory or from equivalent circuit analysis of the connection system for the resonator. The equivalent circuit for coupled resonators has a series resonance

Here, R is the source internal resistance and the frequency is ω(2πf). Vg is called amplitude of the voltage source. Ls and Ld are source and device resonator coils, respectively. And mutual inductance is indicated by *M* =*k L <sup>s</sup>L <sup>d</sup>* . Each coil has a series capacitor that forms a resonator. Rs and Rd resistors denote unwanted resistance (including ohmic and radiation losses) in the coil and the resonance capacitor for each resonator. AC load resistance is shown as RL.

The yield for this circuit can be determined from the ratio of transmitted power for load resistance to the maximum available power of the source, while resonator system is strongly

loss (the reduction of R).

to reduce their interactions with foreign objects.

52 Wireless Power Transfer - Fundamentals and Technologies

circuit structure as shown in **Figure 2**.

**Figure 2.** Equivalent circuit of coupled resonator system.

coupled.

$$
\eta\_{\rm opt} = \frac{U^2}{\left(\mathbb{1} + \sqrt{\mathbb{1} + U^2}\right)^2} \tag{6}
$$

Using Eq. (6), the maximum power transmission efficiency can be depicted as in **Figure 3**.

Systems with large values of U can achieve highly efficient energy transfer. Best wireless energy transfer system efficiency can be possible by determining the performance factors of the system, such as U which is depend on magnetic coupling coefficient k, and quality factors of the Qs (source) and Qd (device).

For certain applications, the resonator quality factor and magnetic coupling between resona‐ tors are used to determine the best possible performance for the system.

For wireless power transfer, it can be seen in Eq. (4) that the magnetic coupling coefficient and quality factor of the significance is greater. The magnetic coupling coefficient represents the magnetic flux connection between the source, while the device resonator is a dimensionless parameter with a value between 0 (unconnected) and 1 (complete flux linked). Conventional induction-based wireless power transmission systems (such as electric toothbrushes) have a high coupling value and a close range, aligned between the source and the device. Eq. (4) indicates that a high-quality resonator is more efficient than a traditional induction system. More importantly, it becomes possible to work efficiently at low coupling values. Also, for this reason, the need for precise positioning between the source and the device is eliminated. Unfortunately, the biggest drawback of the high quality factor is that the capacitor's peak voltage is too high. The relationship between the peak value of the capacitor voltage and the quality is shown in Eq. (7) [17].

(7)

**Figure 3.** Depending on the U function, the optimum efficiency graph of energy transfer.

#### **2.1. Magnetic coupling circuit**

$$V\_1 = I\_1 \left( R + jL\_1 \alpha + \left( \frac{\mathfrak{l}}{j\alpha \text{C}} \right) \right) - I\_2 \left( jL\_n \alpha \right) \tag{8}$$

$$\mathbf{O} = I\_2 \left( jL\_2 \alpha + \left( \frac{\mathbf{1}}{j\alpha C} \right) + Z\_0 + R \right) - I\_1 \left( jL\_n \alpha \right) \tag{9}$$

$$I\_2 \left( jL\_2 \alpha + \left( \frac{\mathfrak{l}}{j\alpha C} \right) + Z\_\alpha + R \right) = I\_1 \left( jL\_m \alpha \right) \tag{10}$$

$$I\_2 = I\_1 \left(\frac{jL\_n\alpha}{jL\_2\alpha + \left(\frac{1}{j\alpha C}\right) + Z\_0 + R}\right) \tag{11}$$

$$V\_1 = I\_1 \left( R + jL\_1 \alpha + \left( \frac{1}{j\alpha C} \right) \right) - I\_1 \left( \frac{jL\_n \alpha}{jL\_2 \alpha + \left( \frac{1}{j\alpha C} \right) + Z\_0 + R} \right) \left( jL\_n \alpha \right) \tag{12}$$

Wireless Power Transfer by Using Magnetically Coupled Resonators http://dx.doi.org/10.5772/64031 55

$$V\_t = I\_t \left[ \left( R + jL\_t o + \left( \frac{\mathbf{1}}{joC} \right) \right) - \left( \frac{j^2 L\_n{}^2 o^2}{jL\_2 o + \left( \frac{\mathbf{1}}{joC} \right) + Z\_o + R} \right) \right] \tag{13}$$

voltage is too high. The relationship between the peak value of the capacitor voltage and the

*s*

(7)

(8)

(10)

*Cpeak <sup>V</sup> V Q <sup>π</sup>* <sup>=</sup> <sup>2</sup>

**Figure 3.** Depending on the U function, the optimum efficiency graph of energy transfer.

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quality is shown in Eq. (7) [17].

54 Wireless Power Transfer - Fundamentals and Technologies

**2.1. Magnetic coupling circuit**

$$Z\_{Eq} = Z\_t = R + jL\_t \alpha + \left(\frac{\mathbf{1}}{j\alpha C}\right) + \left(\frac{L\_n^{-2}\alpha^2}{jL\_2\alpha + \left(\frac{\mathbf{1}}{j\alpha C}\right) + Z\_0 + R}\right) \tag{14}$$

$$Z\_{Eq} = R + jL\_{\text{t}}\alpha + \left(\frac{1}{j\alpha C}\right) + \left(\frac{L\_{\text{m}}^{\ast^2}\alpha^2}{jL\_{\text{2}}\alpha + \left(\frac{1}{j\alpha C}\right) + Z\_0 + R}\right) + jL\_{\text{m}}\alpha - jL\_{\text{m}}\alpha \tag{15}$$

$$Z\_{Eq} = R + \left(\frac{\mathfrak{l}}{j\alpha C}\right) + j(L\_1 - L\_n)\alpha + \frac{-j^2 L\_n{}^2 \alpha^2 + j^2 L\_n L\_2 \alpha^2 + j L\_n \alpha (Z\_0 + R) + j L\_n \left(\frac{\mathfrak{l}}{j\alpha C}\right) \alpha}{j L\_2 \alpha + \left(\frac{\mathfrak{l}}{j\alpha C}\right) + Z\_0 + R} \tag{16}$$

$$Z\_{Eq} = R + \left(\frac{\mathfrak{l}}{joC}\right) + j(L\_1 - L\_n)o + \frac{\left(jL\_n o\right)\left(j(L\_2 - L\_n)o + \left(\frac{\mathfrak{l}}{joC}\right) + Z\_0 + R\right)}{jL\_2o + \left(\frac{\mathfrak{l}}{joC}\right) + Z\_0 + R} \tag{17}$$

$$Z\_{Eq} = R + \left(\frac{1}{j\alpha C}\right) + j(L\_1 - L\_n)\alpha o + \frac{1}{jL\_2\alpha o + \left(\frac{1}{j\alpha C}\right) + Z\_0 + R + jL\_n\alpha o - jL\_n\alpha o}\tag{18}$$

$$\frac{1}{(jL\_n\alpha)(j(L\_2 - L\_n)\alpha o + \left(\frac{1}{j\alpha C}\right) + Z\_0 + R)}\tag{19}$$

$$Z\_{Eq} = R + \left(\frac{1}{j\alpha C}\right) + j(L\_1 - L\_n)\alpha + \frac{1}{jL\_n\alpha} + \frac{1}{j(L\_2 - L\_n)\alpha + (1 \wedge j\alpha C) + Z\_0 + R} \tag{19}$$

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

**Figure 4.** Magnetic coupling circuit.

#### **2.2. Efficiency equation**

$$\eta = \frac{P\_{\text{out}}}{P\_{in}} = \frac{I\_{\text{out}}^2 Z\_{\text{out}}}{I\_{in}^2 Z\_{Eq}} \tag{20}$$

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

$$\frac{I\_{\text{out}}}{I\_{\text{in}}} = \frac{jL\_{\text{in}}\alpha}{jL\_{2}\alpha + \left(\frac{1}{j\alpha C}\right) + Z\_{\text{0}} + R} \tag{21}$$

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

$$\eta = \frac{jL\_n o}{jL\_2 o + \left(\frac{1}{joC}\right) + Z\_o + R})^2 \frac{Z\_o}{\left(\frac{1}{R + jL\_1 o + \left(\frac{1}{joC}\right)} + \left(\frac{L\_n{o}^2 o^2}{jL\_2 o + \left(\frac{1}{joC}\right) + Z\_o + R}\right)\right)}\tag{22}$$

At a given resonant frequency, the conditions for system efficiency are defined for three states, defined by Eqs. (23), (24), and (25).

Wireless Power Transfer by Using Magnetically Coupled Resonators http://dx.doi.org/10.5772/64031 57

$$L\_m^2 = \frac{Z\_0^2 - R^2}{\alpha\_0^2} \tag{23}$$

$$L\_m^2 > \frac{Z\_0^2 - R^2}{\alpha\_0^2} \tag{24}$$

$$L\_m^2 < \frac{Z\_0^2 - R^2}{\alpha\_0^2} \tag{25}$$

Eq. (23) describes the maximum efficiency condition, while Eq. (24) represents the double resonance frequency condition. Eq.(25) describes the condition of the system with a singleresonant frequency at low efficiency level [19].
