**2. Nonresonant magnetic coupling of the WET system**

In the case of nonresonant magnetic coupling, the system acts as an air-cored transformer with relatively low mutual inductance. Due to high supplying frequency, the primary and the secondary leakage reactance cause a significant voltage drop, which limits amount of power delivered to the load. Equivalent circuit composed for this type of coupling is shown in **Figure 1**.

**Figure 1.** Equivalent circuit of the inductively coupled nonresonant WET system.

longchargingtimes.These reasons showhownecessaryitis tosolve anissueofbatterycharging [1]. In general, charging systems can be classified as wired and these could be replaced with

Wireless power transfer and issues relevant to wireless charging of various types of electronic devices are still important and emerging trends in electrical engineering [2‒4]. Regarding of wireless power transfer, each application requires several self-specific operational properties or transfer characteristics. The most important of them are transmitting efficiency, value of the output power, mutual position between the transmitter and the receiver, and their geometric dimensions. One possible way how to influence these parameters lies in the main circuit configuration of a compensation network of the wireless energy transfer (WET) system. Inductively coupled power-transfer systems have been suited for a wide spectrum of consumer applications including electromobility (battery charging of vehicles). Even high system efficiency has been achieved (approximately 70%); however, the restriction to close range, typically shorter than 30% of the coil diameter, is the main limitation for perspective usage in

It the beginning of 2000s, a team of researchers under charge of Prof. Soljacic introduced a new concept of strongly coupled magnetic resonance for wireless energy transfer systems [7]. In comparison with an inductively coupled system, the coils with resonant inductive coupling have several significant advantages. The most important of them are as follows: the coupling may be very efficient even at large transmission distances, it has low environmental impact, and can be variously tuned/configured based on the requirements of target application. The solution of the WET system based on the magnetic-resonance coupling represents constantly investigated phenomena. Most of the analyses are based on the pure theory of physical interactions, which are unable to provide clear and consistent overview of knowledge for researchers in the field of electrical engineering [8, 9]. Based on this, it can be said that some important relationships between system efficiency, amount of transmitted power, and transmitting distance for magnetic-resonance coupled WET systems need to be introduced.

This chapter describes a simple equivalent circuit model for magnetic-coupled and resonant magnetic-coupled WET systems. The resonant coupling is further described in details, whereby all key system equations for various configurations of a resonant compensation network with pure resistive load are derived. From the component design point of view, it is also important to have knowledge about values of voltage and current in the main circuit. In the last part of the chapter, the general recommendations for practical use of each variant are

In the case of nonresonant magnetic coupling, the system acts as an air-cored transformer with relatively low mutual inductance. Due to high supplying frequency, the primary and the secondary leakage reactance cause a significant voltage drop, which limits amount of power

**2. Nonresonant magnetic coupling of the WET system**

wireless (inductive) systems.

28 Wireless Power Transfer - Fundamentals and Technologies

provided.

high-efficiency or high-performance systems [5, 6].

The circuit can be completely described by the system of loop currents calculated by Eq. (1):

$$
\begin{bmatrix} \dot{U}\_1 \\ 0 \end{bmatrix} = \begin{bmatrix} R\_1 + joL\_1 & -joM \\ -joM & R\_2 + R\_L + joL\_2 \end{bmatrix} \begin{bmatrix} \dot{I}\_1 \\ \dot{I}\_2 \end{bmatrix} \tag{1}
$$

The parameters *R*1 and *R*2 represent parasitic resistances of transmitting and receiving coils with self-inductances *L*1 and *L*2, *M* forms the mutual inductance between respective coils, and *R*L is the load (purely resistive). With regard to the high operational frequencies, the switched power source with square-wave voltage should be used to supply the system. In this case, Eq. (1) needs to be rewritten in Eq. (1) to consider selected frequency spectrum.

$$
\begin{bmatrix}
\frac{4}{\pi} \sum\_{k=1,3,5,\dots}^{\alpha} \frac{\dot{U}\_1}{k} \\
0
\end{bmatrix} = \begin{bmatrix}
\sum\_{k=1,3,5,\dots}^{\alpha} R\_1 + jk\alpha L\_1 & \sum\_{k=1,3,5,\dots}^{\alpha} - jk\alpha M \\
\sum\_{k=1,3,5,\dots}^{\alpha} - jk\alpha M & \sum\_{k=1,3,5,\dots}^{\alpha} R\_2 + R\_L + jk\alpha L\_2
\end{bmatrix} \begin{bmatrix}
\dot{I}\_1 \\
\dot{I}\_2
\end{bmatrix} \tag{2}
$$

On the other hand, the uncompensated leakage reactance forms in the circuit with a very high impedance even for a lower frequency component. This impedance causes a significant voltage drop and therefore the contribution of this harmonic component is very low. The system can be therefore analyzed only for the fundamental wave while keeping relatively good accuracy. The winding currents derived for the fundamental wave are provided in Eqs. (3) and (4):

$$\dot{I}\_1 = \frac{R\_2 + R\_L + jo\omega L\_2}{(R\_1 + joL\_1).(R\_2 + R\_L + joL\_2) + (oM)^2} \dot{U}\_1 \tag{3}$$

$$\dot{I}\_2 = \frac{-j\alpha M}{(R\_1 + j\alpha L\_1).(R\_2 + R\_L + j\alpha L\_2) + (\alpha M)^2} \dot{U}\_1 \tag{4}$$

The input power can be counted from the complex power on the primary side (5):

$$P\_{\mathcal{N}} = \Re \left\{ \dot{U}\_1 \dot{I}\_1^\* \right\} \tag{5}$$

The secondary side average power is then found using the loading resistance and the current flowing through it (Eq. (6)):

$$P\_{OUT} = R\_L \left| \dot{I}\_2 \right|^2 = R\_L \left| \frac{-j\alpha M}{(R\_1 + j\alpha L\_1)(R\_2 + R\_L + j\alpha L\_2) + (\alpha M)^2} \dot{U}\_1 \right|^2 \tag{6}$$

The transmitting efficiency (7) is further given by the ratio of the output power (6) and the input power (5):

$$\eta = \frac{P\_{OUT}}{\left| \dot{S}\_{N\uparrow} \right| \cos(\phi)} = \left| \frac{R\_z \rho \dot{\phi}^2 M^2}{\left[ (R\_1 + joL\_1)(R\_2 + R\_L + joL\_2) + (oM)^2 \right] \left( R\_2 + R\_L + joL\_2 \right) \cos(\phi)} \right| \tag{7}$$

where

$$\cos(\varphi) = \frac{\Re\left(\dot{S}\_{N}\right)}{\left|\dot{S}\_{N}\right|} = \frac{\Re\left(\frac{R\_{2} + R\_{L} + joL\_{2}}{(R\_{1} + joL\_{1})(R\_{2} + R\_{L} + joL\_{2}) + (oM)^{2}}\dot{U}^{2}\_{1}\right)}{\left|\frac{R\_{2} + R\_{L} + joL\_{2}}{(R\_{1} + joL\_{1})(R\_{2} + R\_{L} + joL\_{2}) + (oM)^{2}}\dot{U}^{2}\_{1}\right|}\tag{8}$$

The simulation parameters are taken from two constructed experimental prototypes of coupling coils, which are also identical. Their values are therefore the same on the primary as well as on the secondary side. The self-inductance equals *L*1 = *L*2 = 127.4 μH, parasitic resistance *R*1 = *R*2 = 0.447 Ω, and the load *R*L = 8 Ω. The analyzed frequency range (*f*range = 290‒340 kHz) is set according to the resonant frequency obtained from the resonant magnetic coupling discussed in the following chapter. The used compensation capacitors also have the same capacities *C*1 = *C*2 = 2 nF. In the simulation, the minimal mutual inductance *M*min =1 μH corresponds to the transmitting distance of 50 cm, whereas its maximal value *M*max = 11 μH represents the distance of 25 cm. This transmitting range is selected based on the application area, i.e., +battery charging of electric vehicles.

The results from simulation of the inductively coupled WET system plotted against frequency and mutual inductance are shown in **Figures 2**–**4**. The waveforms of the input and output powers are shown in **Figure 2**, the source and load current are shown in **Figure 3**, and the efficiency calculated using Eq. (7) together with a secondary induced voltage *U*L2 are shown in **Figure 4**.

**Figure 2.** Input power (left) and output power (right) dependency on nonresonant coupling.

2 1 2 1 12 2 ( ).( )( ) *L j M I U R jL R R jL M* w

The input power can be counted from the complex power on the primary side (5):

. . ( ).( )( ) *OUT L L*

w

*IN L L*

*S R jL R R jL M R R jL*

( ).( )( ) cos( )

w

w

*IN L IN L*

*j M P RI R U*

 ww

{ } \*

The secondary side average power is then found using the loading resistance and the current

2 2 1 1 12 2

The transmitting efficiency (7) is further given by the ratio of the output power (6) and the

1 12 2 2 2 .cos( ) ( ).( )( ). .cos( )

( ) 2 2 2

*R R jL U S R jL R R jL M*

& &

1 12 2

1 12 2

The simulation parameters are taken from two constructed experimental prototypes of coupling coils, which are also identical. Their values are therefore the same on the primary as well as on the secondary side. The self-inductance equals *L*1 = *L*2 = 127.4 μH, parasitic resistance *R*1 = *R*2 = 0.447 Ω, and the load *R*L = 8 Ω. The analyzed frequency range (*f*range = 290‒340 kHz) is set according to the resonant frequency obtained from the resonant magnetic coupling discussed in the following chapter. The used compensation capacitors also have the same capacities *C*1 = *C*2 = 2 nF. In the simulation, the minimal mutual inductance *M*min =1 μH corresponds to the transmitting distance of 50 cm, whereas its maximal value *M*max = 11 μH represents the distance of 25 cm. This transmitting range is selected based on the application

*S R R jL U R jL R R jL M*

& &

æ ö + + Âç ÷ Â è ø + ++ + = = + +

( ).( )( )

+ ++ +

*L*

*L*

 ww

*L*

 ww

2 2

w

= = é ù + ++ + ++ ë û & (7)

2

2 2 2

w

 ww

w

 ww

& & (6)

*R jL R R jL M* w

& & (4)

1 1 . *P UI IN* = Â & & (5)

2

( )

 w

2 1

2 1

 j

(8)

w

2

*OUT Z*

w

*P R M*

flowing through it (Eq. (6)):

30 Wireless Power Transfer - Fundamentals and Technologies

j

j

area, i.e., +battery charging of electric vehicles.

input power (5):

h

where

**Figure 3.** Source current (left) and load current (right) dependency on nonresonant coupling.

**Figure 4.** Efficiency dependency on nonresonant coupling.

As it can be seen from **Figure 4** (right), the transmitting efficiency mainly varies with given mutual inductance. The system is practically independent of frequency change even for the whole considered frequency range.

#### **3. Series–series resonant coupling of the WET system**

For series–series compensation, the capacitor is connected in series with transmitting and receiving coils (**Figure 5**).

**Figure 5.** Equivalent circuit of the series‒series compensated WET system.

Mathematical model of series–series compensation is, from the point of view of complexity, much easier than other compensation types. Using the methodology of loop currents, the impedance matrix of this configuration can be expressed as follows (11):

$$\left[R\_1 + j\left(\alpha L\_1 - \frac{1}{\alpha C\_1}\right)\right] \dot{I}\_{1\\_ \text{LS}} - j\alpha M. \dot{I}\_{2\\_ \text{LS}} = \dot{U}\_{1\\_ \text{LS}} \tag{9}$$

$$-j\alpha M \, \dot{I}\_{1\\_SS} + \left[ R\_2 + R\_L + j \left( \alpha L\_2 - \frac{1}{\alpha C\_2} \right) \right] \dot{I}\_{2\\_SS} = 0 \tag{10}$$

$$
\begin{bmatrix}
\dot{U}\_{1\_{-\rm SS}} \\
0
\end{bmatrix} = \begin{bmatrix}
R\_1 + j \left(ooL\_1 - \frac{1}{ooC\_1}\right) & -jooM \\
& -jooM & -R\_2 + R\_L + j \left(ooL\_2 - \frac{1}{ooC\_2}\right) \\
& & -j \left(ooL\_2 - \frac{1}{ooC\_2}\right)
\end{bmatrix} \begin{bmatrix}
\dot{I}\_{1\_{-\rm SS}} \\
\dot{I}\_{2\_{-\rm SS}}
\end{bmatrix} \tag{11}
$$

From (11), the formula for the current of the transmitting and receiving parts in the complex form as (12) and (13) can be derived:

$$\dot{I}\_{1} = \dot{I}\_{1\\_SS} = \frac{R\_2 + R\_L + j\left(\alpha L\_2 - \frac{1}{\alpha C\_2}\right)}{\left[R\_1 + j\left(\alpha L\_1 - \frac{1}{\alpha C\_1}\right)\right] \left[R\_2 + R\_L + j\left(\alpha L\_2 - \frac{1}{\alpha C\_2}\right)\right] + \left(\alpha M\right)^2} \tag{12}$$

Analysis of Wireless Power System Efficiency in Dependency on Configuration of Resonant Tank http://dx.doi.org/10.5772/62998 33

$$\dot{I}\_{2} = \dot{I}\_{2\_{-\rm{LS}}} = \frac{-j\alpha M}{\left[R\_{1} + j\left(\alpha L\_{1} - \frac{1}{\alpha \mathcal{C}\_{1}}\right)\right] \left[R\_{2} + R\_{L} + j\left(\alpha L\_{2} - \frac{1}{\alpha \mathcal{C}\_{2}}\right)\right] + \left(\alpha M\right)^{2}} \tag{13}$$

If the circuit is supplied by harmonic voltage with a frequency equal to the resonant fre‐ quency, then the value of the capacitive and inductive parts of the complex impedance will be the same and can be subtracted. Then, for the previous formulas (14) and (15), the follow‐ ing conditions are valid:

**3. Series–series resonant coupling of the WET system**

**Figure 5.** Equivalent circuit of the series‒series compensated WET system.

w

1 1

w

w

w

w

form as (12) and (13) can be derived:

impedance matrix of this configuration can be expressed as follows (11):

1

1

w

é ù æ ö ê ú +- - = ç ÷ ë û è ø

w

receiving coils (**Figure 5**).

32 Wireless Power Transfer - Fundamentals and Technologies

For series–series compensation, the capacitor is connected in series with transmitting and

Mathematical model of series–series compensation is, from the point of view of complexity, much easier than other compensation types. Using the methodology of loop currents, the

1 1 1\_ 2\_ 1\_

1 . . *R j L I j MI U SS SS SS C*

1\_ 2 2 2 \_

 w

é ù æ ö - + ++ - = ê ú ç ÷ ë û è ø

1 . . 0 *SS L SS j MI R R j L I C*

1\_ 1 1\_

2 2

*RR jL C I I U*

*L*

1 1\_ 1\_

11 2 2

è ø = = é æö æ ö ù é ù ê + - ++ - + ç÷ ç ÷ ú ê ú ë èø è ø û ë û

w

é ù æ ö ê ú +- - ç ÷ é ù é ù è ø ê ú = ê ú ë û æ ö ê ú ë û - ++ - ç ÷ ë û è ø & &

. 0 1 *SS SS*

*RjL j M U C I*

 w

2

& & (10)

w

 w

> w

> > 2

ww

 w

1

1 1 . ( )

w

æ ö ++ - ç ÷

& & & (12)

2 2

*L*

*jM R R j L C*

From (11), the formula for the current of the transmitting and receiving parts in the complex

1 2

*SS SS L*

w

*RjL RR jL M C C*

& && (9)

2 \_

*I*

*SS*

2

& (11)

2

w

$$\dot{I}\_{1} = \dot{I}\_{1\_{-} \text{ SS}} = \frac{R\_{2} + R\_{L}}{R\_{1}R\_{2} + R\_{1}R\_{L} + \left(\alpha M\right)^{2}} \dot{U}\_{1\_{-} \text{ SS}} \tag{14}$$

$$\dot{I}\_2 = \dot{I}\_{2\\_SS} = \frac{-j\alpha M}{R\_1 R\_2 + R\_1 R\_L + (\alpha M)^2} \dot{U}\_{1\\_SS} \tag{15}$$

Equations (14) and (15) show that the circuit during resonance has only resistive characteristics, and circuit currents are given just by the parasitic resistances of coils, load resistance, and supply voltage. The input and output powers of the circuit with –series‒series compensation can be expressed as Eqs. (16) and (17). The graphical interpretation of Eq. (17), as plotted in **Figure 6**, shows that the operation of the system with this kind of compensation should be excluded from resonant point. In other case, for low values of *M* (high distance of coils), the input power, as well as the output power (output current), rises too sharply, which might destroy the whole system.

$$\dot{S}\_{\text{IN\\_SS}} = \dot{I}\_1 \dot{U}\_{1\\_\text{SS}} = \frac{R\_2 + R\_L + j\alpha L\_2}{\left[R\_1 + j\left(\alpha L\_1 - \frac{1}{\alpha C\_1}\right)\right] \left[R\_2 + R\_L + j\left(\alpha L\_2 - \frac{1}{\alpha C\_2}\right)\right] + (\alpha M)^2} \tag{16}$$

$$P\_{OUT\\_SS} = R\_L \left| I\_z \right|^2 = R\_L \left| \frac{-j\alpha M}{\left[ \left[ R\_1 + j \left( \alpha L\_1 - \frac{1}{\alpha C\_1} \right) \right] \left[ R\_2 + R\_L + j \left( \alpha L\_2 - \frac{1}{\alpha C\_2} \right) \right] + (\alpha M)^2 \right]} \dot{U}\_{1\\_SS} \right|^2} \tag{17}$$

Dependence of efficiency for the series‒series compensated circuit in the frequency domain can be expressed using Eq. (18):

$$\eta = \frac{\dot{P}\_{\text{OUT\\_IS}}}{S\_{\text{IN\\_IS}} \cos(\rho)}$$

$$\eta = \frac{R\_L \dot{o} a^2 M^2}{\left[ \left[ R\_1 + j \left( a \delta L\_1 - \frac{1}{a \delta C\_1} \right) \right] \left[ R\_2 + R\_L + j \left( a \delta L\_2 - \frac{1}{a \delta C\_2} \right) \right] + (a \delta M)^2 \right] \left( R\_2 + R\_L + j \alpha L\_2 \right) \cos(\rho)} \tag{18}$$

where

$$\cos(\varphi) = \frac{\Re\left(\bar{S}\_{\mathcal{M}}\right)}{\left|\bar{S}\_{\mathcal{M}}\right|} = \frac{\Re\left(\frac{R\_{\star} + R\_{\downarrow} + joL\_{\downarrow}}{\left[\left(R\_{\star} + j\left(oL\_{1} - \frac{1}{oC\_{1}}\right)\right)\right]\left[\left(R\_{\star} + R\_{\downarrow} + j\left(oL\_{2} - \frac{1}{oC\_{2}}\right)\right] + \left(ooM\right)^{2}\right]}{\left[\left(R\_{\star} + R\_{\downarrow} + joL\_{\downarrow}\right)\right]\left[\left(R\_{\star} + R\_{\downarrow} + j\left(oL\_{2} - \frac{1}{oC\_{2}}\right)\right)\right]\left(oM\right)^{2}}\right]}\tag{19}$$

**Figure 6.** Input power (left) and output power (right) dependency on series‒series coupling.

The highest efficiency can be achieved at the resonant frequency. Its value is above 90% when higher values of mutual inductance are considered. The main disadvantage of high efficiency achievement is that for a given value of *M*, the transmitted power is too low (**Figure 6**, right). This fact is crucial when simultaneously high efficiency together with high power transfer is required. The solution is to operate the system at the frequency which is given by Eq. (20).

$$\left[R\_{\parallel} + j\left(\alpha L\_{\parallel} - \frac{1}{\alpha C\_{\parallel}}\right)\right] \dot{I}\_{\parallel \\_ SP} - j\alpha \mathcal{M}. \dot{I}\_{\uparrow \\_ SP} = \dot{U}\_{\downarrow \\_ SP} \tag{20}$$

At such frequency, with value depending on the value of mutual inductance (distance between the transmitting and receiving coils), it is possible to achieve a quite high value of the system efficiency together with peak power transfer. For example, when the system has *M* = 5 μH, which is relevant to 40 cm of distance between coils, then almost 85% of the system efficiency can be achieved at the peak power that is transferred to the load. As it can be further seen in **Figure 7** (right), the value of efficiency naturally depends on the value of the output load. Anyway, it is still possible to achieve an acceptable value of system efficiency even at the sharp change of the output load. Such behavior is mostly welcomed for perspective wireless charging systems, where high transmitting distance, high efficiency, high transferred power, and variable load are required.

( )

wj

1\_ 2

&

*SS*

1\_ 2

&

*SS*

(18)

(19)

(20)

2

2 2 2

 w

ww

*R R jL U*

2 2 2

 w

ww

*R R jL U*

1 1 . ( ). .cos( )

 w

1 2

w

w

æ ö ç ÷

*L*

*L*

+ + Â é æö æ ö ù é ù ê + - ++ - + ç÷ ç ÷ ú ê ú Â ç ÷

è ø ë èø è ø û ë û = =

é æö æ ö ù é ù ê + - ++ - + ç÷ ç ÷ ú ê ú ë èø è ø û ë û

The highest efficiency can be achieved at the resonant frequency. Its value is above 90% when higher values of mutual inductance are considered. The main disadvantage of high efficiency achievement is that for a given value of *M*, the transmitted power is too low (**Figure 6**, right). This fact is crucial when simultaneously high efficiency together with high power transfer is required. The solution is to operate the system at the frequency which is given by Eq. (20).

At such frequency, with value depending on the value of mutual inductance (distance between the transmitting and receiving coils), it is possible to achieve a quite high value of the system

*L*

+ +

*RjL RR jL M C C*

1 1 . ( )

1 1 . ( )

1 2

*L*

*L L*

2 2

11 2 2 2 2

11 2 2

w

w

**Figure 6.** Input power (left) and output power (right) dependency on series‒series coupling.

*RjL RR jL M S C C*

11 2 2

*R j L R R j L M R R jL C C*

w

*L*

*R M*

 w

w

1 2

w

w

= ì ü ï ï é æö æ ö ù é ù í ý ê + - ++ - + ++ ç÷ ç ÷ ú ê ú ï ï î þ ë èø è ø û ë û

\_

*OUT SS*

.cos( )

w

( )

& &

*S*

*IN IN*

j

w

34 Wireless Power Transfer - Fundamentals and Technologies

\_

*P S*

& &

*IN SS*

h

where

cos( )

j

=

**Figure 7.** Dependency of efficiency on the series‒series coupling (left) and on the output load change at *M* = 11 μH (right).

From the practical point of view, during the design process of the system, it is necessary to know the voltage waveform at compensation capacitors because these components are the most critical. Figures 8 and 9 show voltage dependency for each system component. It can be seen that even for a very low value of the supply voltage (in this case 30 V), the peak voltage at each component multiplies several times. Situation is most critical when the system operates at resonant frequency and also when a low value of mutual inductance is presented. If the input voltage rises, then naturally each particular voltage rises correspondingly too. The selection of the proper capacitor structure and configuration therefore means the most difficult issue. Thus, the previous recommendation for system operation at border resonant frequencies (19) also gives advantages from the system component design point of view. The same is valid

**Figure 8.** Dependency of voltage on the primary compensation capacitor *C*1 (left) and on the primary coil *L*1 (right) for series‒series coupling.

for the wire selection of the transmitting and receiving coils, when the source and load current show similar dependency, as the previous circuit variables (**Figure 10**).

**Figure 9.** Dependency of voltage on the secondary compensation capacitor *C*2 (left) and on the secondary coil *L*2 (right) for series‒series coupling.

**Figure 10.** Source current (left) and load current (right) dependency on series‒series coupling.

#### **4. Series–parallel resonant coupling of the WET system**

Series‒parallel compensation means that one capacitor is connected in series to the transmit‐ ting coil and the other one is connected in parallel to the receiving coil, see **Figure 11**.

Using the method of loop currents, the impedance matrix of series–parallel compensated circuit can be expressed as follows (24):

$$-j\alpha \mathcal{M} \dot{I}\_{1\\_SP} + \left[ R\_\odot + j \left( \alpha \mathcal{L}\_\odot - \frac{1}{\alpha \mathcal{C}\_\odot} \right) \right] \dot{I}\_{2\\_SP} + j \frac{1}{\alpha \mathcal{C}\_\odot} \dot{I}\_{3\\_SP} = 0 \tag{21}$$

Analysis of Wireless Power System Efficiency in Dependency on Configuration of Resonant Tank http://dx.doi.org/10.5772/62998 37

$$Z\_1 = R\_1 + j\left(\alpha \mathcal{L}\_1 - \frac{1}{\alpha \mathcal{C}\_1}\right)$$

$$Z\_2 = R\_2 + j\left(\alpha \mathcal{L}\_2 - \frac{1}{\alpha \mathcal{C}\_2}\right)$$

$$j\frac{1}{\alpha \mathcal{C}\_2} \dot{J}\_{2\\_Sp} + \left[R\_L - j\frac{1}{\alpha \mathcal{C}\_2}\right] \dot{J}\_{3\\_Sp} = 0\tag{22}$$

$$
\begin{bmatrix}
\dot{U}\_{\text{\tiny\\_sp}} \\
0 \\
0
\end{bmatrix} = \begin{bmatrix}
R\_{\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\tiny\text{\text}}}}}}}}}}}}
\end\text{\}}
}}
\end\|}
}}
\,\text{\text{\text{\tiny\text{\text{\textdegree}}}}
\,\text{\text{\textquotedblleft}}
\,\text{\textquotedblright}}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblleft}
\,\text{\textquotedblright}}
\,\text{\textquotedblright}
\,\text{\textquotedblright}
\end{bmatrix}
\end{bmatrix}
$$

for the wire selection of the transmitting and receiving coils, when the source and load current

**Figure 9.** Dependency of voltage on the secondary compensation capacitor *C*2 (left) and on the secondary coil *L*2 (right)

Series‒parallel compensation means that one capacitor is connected in series to the transmit‐

Using the method of loop currents, the impedance matrix of series–parallel compensated

(21)

ting coil and the other one is connected in parallel to the receiving coil, see **Figure 11**.

**Figure 10.** Source current (left) and load current (right) dependency on series‒series coupling.

**4. Series–parallel resonant coupling of the WET system**

circuit can be expressed as follows (24):

show similar dependency, as the previous circuit variables (**Figure 10**).

36 Wireless Power Transfer - Fundamentals and Technologies

for series‒series coupling.

$$
\begin{bmatrix} U\_{1\\_SP} \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} R\_1 + j \left( ooL\_1 - \frac{1}{ooC\_1} \right) & -j ooM & 0 \\\\ & -j ooM & R\_2 + j \left( oL\_2 - \frac{1}{ooC\_2} \right) & j \frac{1}{oC\_2} \\\\ 0 & j \frac{1}{ooC\_2} & R\_L - j \frac{1}{oC\_2} \end{bmatrix} \begin{bmatrix} \dot{I}\_{1\\_SP} \\ \dot{I}\_{2\\_SP} \\ \dot{I}\_{3\\_SP} \end{bmatrix} \tag{24}
$$

If we substitute *Z*1 =*R*1 + j(*ωL* 1 − 1 *ωC*1 ) and *Z*2 =*R*2 + j(*ωL* 2 − 1 *ωC*2 ), the circuit currents can be ex‐ pressed as follows:

$$\dot{I}\_{1} = \dot{I}\_{1\_{-},SP} = \frac{R\_{2} + j\frac{1}{o\alpha C\_{2}\left(j o C\_{2}R\_{L} + 1\right)}}{R\_{1}R\_{2} + o^{2}M^{2} + j\frac{R\_{1}}{o\alpha C\_{2}\left(j o C\_{2}R\_{L} + 1\right)}}\dot{U}\_{1\_{-},SP} \tag{25}$$

$$\dot{I}\_2 = \dot{I}\_{2\\_SP} = \frac{jooM}{R\_1R\_2 + o^2M^2 + j\frac{R\_1}{oO\_2\left(jooC\_2R\_L + 1\right)}}\dot{U}\_{1\\_SP} \tag{26}$$

$$\dot{I}\_3 = \dot{I}\_{2\\_SP} - \dot{I}\_{3\\_SP} = \frac{-o\alpha^2 C\_2 R\_L M}{\left(R\_1 R\_2 + o\alpha^2 M^2\right)\left(j o C\_2 R\_L + 1\right) + j \frac{R\_1}{o\alpha C\_2}} \dot{U}\_{1\\_SP} \tag{27}$$

**Figure 11.** Equivalent circuit of the series‒parallel compensated WET system.

Compared with the series‒series compensation, in this case the reactance of capacitor *C*2 in‐ fluences circuit variables when circuit is powered by the resonant frequency because *C*2 is in parallel to the load resistance *R*L. In this case, the power consumption, output power, and efficiency can be calculated in the same way as in the case of series‒series compensation (Eqs. (28)–(31)).

$$\dot{S}\_{N\_{\perp}\text{-}S^{p}} = \dot{I}\_{1}\dot{U}\_{1\\_Sl^{p}} = \frac{Z\_{2} + j\frac{1}{oC\_{2}\left(j o C\_{2}R\_{L} + 1\right)}}{Z\_{1}Z\_{2} + o^{2}M^{2} + j\frac{Z\_{1}}{oC\_{2}\left(j o C\_{2}R\_{L} + 1\right)}}\dot{U}\_{1\\_Sl}^{2} \tag{29}$$

$$P\_{OUT\\_SP} = R\_L \left| I\_4 \right|^2 = \left| \frac{-\dot{\alpha}^2 M^2 R\_L}{\left[ \left( Z\_1 Z\_2 + \alpha^2 M^2 \right) \left( j\alpha C\_2 R\_L + 1 \right) + j \frac{Z\_1}{\alpha C\_2} \right]^2} \dot{U}\_{1\\_SP}^2 \right| \tag{30}$$

$$\begin{aligned} \eta &= \left| \frac{P\_{\text{OUT\\_SP}}}{\bar{S}\_{\text{IN\\_SP}} \cos(\rho)} \right| \\\\ &= \left| \frac{-\alpha^2 M^2 R\_L}{Z\_2 \left( j\alpha C\_2 R\_L + 1 \right) + j \frac{1}{\alpha C\_2} \left| \left[ \left( Z\_1 Z\_2 + \alpha^2 M^2 \right) \left( j\alpha C\_2 R\_L + 1 \right) + j \frac{Z\_1}{\alpha C\_2} \right] \cos(\rho) \right|} \right| \end{aligned} \tag{31}$$

Analysis of Wireless Power System Efficiency in Dependency on Configuration of Resonant Tank http://dx.doi.org/10.5772/62998 39

$$\cos(\rho) = \frac{\Re\left(\dot{S}\_{\mathcal{W}}\right)}{\left|\dot{S}\_{\mathcal{W}}\right|} = \frac{\Re\left(\frac{1}{Z\_1 Z\_2 + \alpha^2 M^2 + j\frac{Z\_1}{\alpha C\_2 \left(j\rho C\_2 R\_L + 1\right)} U\_{1\\_SP}^2}{Z\_2 + j\frac{1}{\alpha C\_2 \left(j\rho C\_2 R\_L + 1\right)}}\right)}{Z\_2 + j\frac{1}{\alpha C\_2 \left(j\rho C\_2 R\_L + 1\right)}}\tag{32}$$

$$\left|\frac{Z\_1 Z\_2 + \alpha^2 M^2 + j\frac{Z\_1}{\alpha C\_2 \left(j\rho C\_2 R\_L + 1\right)}}\dot{U}\_{1\\_SP}^2\right|$$

The graphical interpretations of the mentioned variables are introduced depending on frequency as well as *M* (Figures 12–14). Small change of frequency from resonant frequency causes that power consumption and power delivered to the load are significantly reduced (**Figure 12**). Considering similar dependency for efficiency, it shows no change when frequen‐ cy is being changed (**Figure 13**, left). This is advantageous when power delivery at constant efficiency needs to be managed.

( )( ) 4 3\_ 1\_ 2 2 1 1 2 2

*j M I I U R RR M j C R j C*

& & &

w

= =

38 Wireless Power Transfer - Fundamentals and Technologies

**Figure 11.** Equivalent circuit of the series‒parallel compensated WET system.

(Eqs. (28)–(31)).

1 *SP SP L*

Compared with the series‒series compensation, in this case the reactance of capacitor *C*2 in‐ fluences circuit variables when circuit is powered by the resonant frequency because *C*2 is in parallel to the load resistance *R*L. In this case, the power consumption, output power, and efficiency can be calculated in the same way as in the case of series‒series compensation

2

1 2

2 2 1 2 2

w

*L L*

*Z Z j CR j ZZ M j CR j C C*

w

.

( )

j

\_

*OUT SP*

.cos

\_

&

h

=

*IN SP*

w

*P S*

\_ 1 1\_ 1\_ 2 2 1

( )( )

( ) ( )( ) ( )

2 2

 w

1 1 1 .cos

*L*

2 2

w

*M R*

\_ 4 2 1\_

*OUT SP L SP*

*M R P RI U*

w

1 2 2

*IN SP SP SP*

1 .

*Z j C j CR S IU U Z ZZ M j C j CR* w

+ + = = + +

w

( )

& & & & (29)

1

 w

w

2 2 2 2

w

*Z ZZ M j CR j C*

é ù ê ú + + + ë û

 w

2 2 1

*L*

*L*

1

2 2 1

2 2

 w

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*L*

1

2

 w & (30)

(31)

j

w

+

2 2 2

w

 w

+ + +

2

(28)

w

**Figure 12.** Input power (left) and output power (right) dependency on series‒parallel coupling.

On the other hand, the poor efficiency is the highest disadvantage. Let now consider a different situation, when *M* reaches higher values (reduction of distance between coils). **Figure 13** (right) shows that the efficiency is significantly increased. This means that the best efficiency can be achieved for a short distance, while power transfer to the load is comparable with power transfer for series‒series compensation, see **Figure 14** (left).

Efficiency also depends on the change of the load value. For series‒parallel compensation, the higher efficiency can be achieved at higher values of load resistance (reduction of delivered power). **Figure 14** (right) shows dependency of system efficiency on series‒parallel compen‐ sation at constant *M* = 11 μH during load change. It can be seen that for lower transfer distances (*M* = 11 μH is relevant to 20 cm of mutual distance), the efficiency can be kept at high values even for a wide range of load (**Figure 14**).

**Figure 13.** Dependency of efficiency on series‒parallel coupling (left) and on higher values of mutual inductance (right).

**Figure 14.** Dependency of the output power on higher values of mutual inductance (left) and dependency of efficiency (right) on variable load at constant *M* = 11 μH.

Voltage on capacitor *C*1 (**Figure 15**) in the series‒parallel compensated system also achieves high levels. A decrease or increase in the switching frequency not only reduces the voltage stresses on the capacitor, but also significantly reduces the transmitted power. The same situation is valid for the voltage on coil *L*1.

**Figure 15.** Dependency of voltage on the primary compensation capacitor *C*1 (left) and on the primary coil *L*1 (right) for series‒parallel coupling.

Voltage on capacitor *C*2 (**Figure 16**, left) is the voltage on the resistive load and the receiver coil *L*2 too because they are connected in parallel. In contrast to the voltage on capacitor *C*1, the voltage on capacitor *C*2 is smaller. This voltage is increased when value of *M* is increased too, but only up to a certain value of the mutual inductance, in our case it is 30 μH. The voltage on capacitor *C*2 falls above this value. Capacitor *C*2 does not have to be necessary designed for high voltage stresses, which reduces the overall costs of the system.

**Figure 13.** Dependency of efficiency on series‒parallel coupling (left) and on higher values of mutual inductance

**Figure 14.** Dependency of the output power on higher values of mutual inductance (left) and dependency of efficiency

Voltage on capacitor *C*1 (**Figure 15**) in the series‒parallel compensated system also achieves high levels. A decrease or increase in the switching frequency not only reduces the voltage stresses on the capacitor, but also significantly reduces the transmitted power. The same

**Figure 15.** Dependency of voltage on the primary compensation capacitor *C*1 (left) and on the primary coil *L*1 (right) for

(right).

(right) on variable load at constant *M* = 11 μH.

40 Wireless Power Transfer - Fundamentals and Technologies

situation is valid for the voltage on coil *L*1.

series‒parallel coupling.

**Figure 16.** Dependency of voltage on the secondary compensation capacitor *C*2 (left) and on the secondary coil *L*2 (right) for series‒parallel coupling.

Frequency dependencies of circuit currents are shown in Figures 17 and 18. It can be seen (**Figure 17**, left) that *I*1 decreases with the increase in the mutual inductance and its peak point is shifted to higher frequencies contrary to resonant frequency.

Figures 17 (right) and 18 show the current of the receiving coil *I*2, divided into current of capacitor *C*2 and a load resistance *R*L. Because the value of load resistor is realativelly small, *R*L = 8 Ω, the majority of coil current flows just through load resistance, *I*4. Current *I*3 of the secondary capacitor *C*2 has smaller magnitude. With the increase in the mutual inductance, the maximum of all currents rises.

**Figure 17.** Dependency of current *I*1 (left) and *I*2 (right) on series‒parallel coupling.

**Figure 18.** Dependency of current *I*3 (left) and *I*4 (right) on series‒parallel coupling.

#### **5. Parallel–series resonant coupling of the WET system**

**Figure 19.** Equivalent circuit of the parallel‒series compensated WET system.

Equivalent circuit for parallel‒series compensation is shown in **Figure 19**. Capacitor *C*1 is connected in parallel to the power supply and transmitting coil. The secondary consists of series connection of *L*2–*C*2–*R*L.

With consideration of equivalent circuit shown in **Figure 9**, it is possible to express a system of three linear equations with three unknown parameters (14).

$$
\begin{bmatrix}
\dot{U}\_{1\\_PS} \\
0 \\
0
\end{bmatrix} = \begin{bmatrix}
j\frac{1}{\dot{\alpha}C\_1} & R\_1 + j\alpha L\_1 - j\frac{1}{\dot{\alpha}C\_1} & -j\alpha M & \\
& \begin{bmatrix} 0 & -j\alpha M & \\ & \dot{I}\_2 & -\frac{1}{\dot{\alpha}C\_2} \end{bmatrix} \\
0 & -j\alpha M & R\_2 + R\_L + j\left(\alpha L\_2 - \frac{1}{\dot{\alpha}C\_2}\right) \\
\end{bmatrix} \tag{33}
$$

Equation (32) gives relatively complicated results and therefore some simplifying substitution should be applied: *Z*1 =*R*1 + j(*ωL* 1 − 1 *ωC*1 ) and *Z*2 =*R*2 + *R*L + j(*ωL* 2 − 1 *ωC*2 ). The aimed winding and load currents are determined by Eqs. (33)‒(35):

$$\dot{I}\_1 = \dot{I}\_{1\_{-PS}} = \frac{j\alpha C\_1 (Z\_1 Z\_2 + \alpha^2 M^2)}{Z\_1 Z\_2 + \left(\alpha M\right)^2 + j\frac{Z\_2}{\alpha C\_1}} \dot{U}\_{1\_{-PS}} \tag{34}$$

$$\dot{I}\_{2} = \dot{I}\_{1\\_PS} - \dot{I}\_{2\\_PS} = \frac{j\alpha C\_1 (Z\_1 Z\_2 + \alpha^2 M^2) - Z\_2}{Z\_1 Z\_2 + \left(\alpha M\right)^2 + j\frac{Z\_2}{\alpha C\_1}} \dot{U}\_{1\\_PS} \tag{35}$$

$$\dot{I}\_4 = \dot{I}\_{\text{3\\_PS}} = \frac{j\alpha M}{Z\_1 Z\_2 + \left(\alpha M\right)^2 + j\frac{Z\_2}{\alpha C\_1}} \dot{U}\_{\text{1\\_PS}}\tag{36}$$

**Figure 20.** Input power (left) and output power (right) dependency on series‒parallel coupling.

**Figure 18.** Dependency of current *I*3 (left) and *I*4 (right) on series‒parallel coupling.

42 Wireless Power Transfer - Fundamentals and Technologies

**Figure 19.** Equivalent circuit of the parallel‒series compensated WET system.

of three linear equations with three unknown parameters (14).

1 1

w

w

1 1

w

w

 w

*C C*

series connection of *L*2–*C*2–*R*L.

0

Equivalent circuit for parallel‒series compensation is shown in **Figure 19**. Capacitor *C*1 is connected in parallel to the power supply and transmitting coil. The secondary consists of

With consideration of equivalent circuit shown in **Figure 9**, it is possible to express a system

1 1 2 \_

2 2

 w

> w

*L*

*jM R R j L C*

3 \_

*I*

& &

*PS PS*

(33)

2

w

1\_ 1\_

*PS PS*

é ù ê ú é ù é ù ê ú ê ú = +- - ê ú ê ú

*j R jL j j M I*

1 0

ê ú ê ú ë û ë û æ ö - ++ - ç ÷ ë û è ø

1 1 0

*j j C C U I*

& &

1 1 0 .

 w

**5. Parallel–series resonant coupling of the WET system**

The input and output powers are then found in the same way as from previous analyses (36)– (37).

$$\dot{S}\_{IN\\_PS} = \dot{I}\_1 \dot{U}\_{1\\_PS} = \frac{j\alpha C\_1 (Z\_1 Z\_2 + \alpha^2 M^2)}{Z\_1 Z\_2 + \left(\alpha M\right)^2 + j\frac{Z\_2}{\alpha C\_1}} \dot{U}\_{1\\_PS}^2 \tag{37}$$

$$P\_{OUT\\_SP} = R\_L \cdot \left| \dot{I}\_4 \right|^2 = \left| \frac{-\alpha^2 M^2 R\_L}{\left[ Z\_1 Z\_2 + \left( \alpha M \right)^2 + j \frac{Z\_2}{\alpha C\_1} \right]^2} \dot{U}\_{1\\_SP}^2 \right| \tag{38}$$

where

$$\cos(\rho) = \frac{\Re\left(\dot{S}\_{N}\right)}{\left|\dot{S}\_{N}\right|} = \frac{\left(\frac{j\alpha C\_{1}\left(Z\_{1}Z\_{2} + \alpha^{2}M^{2}\right)}{Z\_{1}Z\_{2} + \left(\alpha M\right)^{2} + j\frac{Z\_{2}}{\alpha C\_{1}}}\dot{U}^{2}\_{1\\_PS}\right)}{\left|\frac{j\alpha C\_{1}\left(Z\_{1}Z\_{2} + \alpha^{2}M^{2}\right)}{Z\_{1}Z\_{2} + \left(\alpha M\right)^{2} + j\frac{Z\_{2}}{\alpha C\_{1}}}\dot{U}^{2}\_{1\\_PS}\right|}\tag{39}$$

The graphical interpretation of Eq. (37) is shown in **Figure 20**. It shows that any frequency deviation from the resonant state reduces the power delivered to the load. However, in this case the system is less sensitive to frequency change as compared to the series‒parallel compensation.

The power delivery to the load for this type of compensation and for the selected operational variables (load, mutual inductance, etc.) is, when compared to the previous types, very poor. Efficiency dependency (**Figure 21**) is comparable to series‒parallel compensation, whereby the higher the value of the mutual inductance is, the higher efficiency can be achieved. Similarly, as for series‒parallel compensation it is expected that the parallel‒series compensation type is more valuable for higher values of mutual inductances, and thus for applications, where distance is not the primary attribution of the WET system.

The advantage of this configuration is the low electrical stress of individual components (Figures 22 and 23).

**Figure 21.** Dependency of efficiency on parallel‒series coupling.

Analysis of Wireless Power System Efficiency in Dependency on Configuration of Resonant Tank http://dx.doi.org/10.5772/62998 45

**Figure 22.** Dependency of voltage on the secondary compensation capacitor *C*2 (left) and on the secondary coil *L*2 (right) for parallel‒series coupling.

#### **6. Parallel–parallel resonant coupling of the WET system**

where

compensation.

(Figures 22 and 23).

( ) ( )

*IN IN*

& &

*S*

distance is not the primary attribution of the WET system.

**Figure 21.** Dependency of efficiency on parallel‒series coupling.

cos( )

44 Wireless Power Transfer - Fundamentals and Technologies

j

1 2

w

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+ + Â è ø = =

w

.

1 2

w

.

( )

w

The graphical interpretation of Eq. (37) is shown in **Figure 20**. It shows that any frequency deviation from the resonant state reduces the power delivered to the load. However, in this case the system is less sensitive to frequency change as compared to the series‒parallel

The power delivery to the load for this type of compensation and for the selected operational variables (load, mutual inductance, etc.) is, when compared to the previous types, very poor. Efficiency dependency (**Figure 21**) is comparable to series‒parallel compensation, whereby the higher the value of the mutual inductance is, the higher efficiency can be achieved. Similarly, as for series‒parallel compensation it is expected that the parallel‒series compensation type is more valuable for higher values of mutual inductances, and thus for applications, where

The advantage of this configuration is the low electrical stress of individual components

*j C ZZ M U Z ZZ M j C*

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( )

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 w

( )

+ + +

 w

2 2 1 12 2 1\_ 2 2

1

&

w

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(39)

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1

w

Equivalent circuit of the system is shown in **Figure 23**. Both compensation capacitors are connected in parallel to the transmitting and receiving coils. Unlike the previous topologies, this model forms the system of four linear equations with four unknown parameters (26):

**Figure 23.** Equivalent circuit of the parallel‒parallel compensated WET system.

$$
\begin{bmatrix}
\dot{U}\_{1\\_PS} \\
0 \\
0 \\
0
\end{bmatrix} = \begin{bmatrix}
j\frac{1}{\alpha C\_1} & R\_1 + j\alpha L\_1 - j\frac{1}{\alpha C\_1} & -j\alpha M & 0 \\
0 & -j\alpha M & R\_2 + j\alpha L\_2 - j\frac{1}{\alpha C\_2} & j\frac{1}{\alpha C\_2} \\
& & \\
0 & 0 & j\frac{1}{\alpha C\_2} & R\_L - j\frac{1}{\alpha C\_2}
\end{bmatrix} \begin{bmatrix}
\dot{I}\_{1\\_PS} \\
\dot{I}\_{2\\_P} \\
\dot{I}\_{3\\_P} \\
\dot{I}\_{4\\_P}
\end{bmatrix} \tag{40}
$$

After substitution *Z*1 =*R*1 + j(*ωL* 1 − 1 *ωC*1 ), *Z*2 =*R*2 + j(*ωL* 2 − 1 *ωC*2 ), and *Z*3 =*RL* −j( 1 *ωC*2 ), the most important circuit variables can be obtained as follows:

$$\dot{I}\_{1} = \dot{I}\_{1\\_PP} = \left[ j\alpha C\_1 + \frac{1}{\left(Z\_1 + j\frac{1}{\alpha C\_1}\right) + \frac{\alpha^4 C\_2^2}{\alpha^2 C\_2^2} \frac{Z\_3 M^2}{Z\_2 Z\_3 + 1}}\right] \dot{U}\_{1\\_PP} \tag{41}$$

$$\dot{I}\_{6} = \dot{I}\_{4\_{-}^{-}P^{p}} = \frac{-\frac{M}{Z\_{3}C\_{2}}}{\left(Z\_{1} + j\frac{1}{\alpha C\_{1}}\right)\left(\frac{1}{\alpha^{2}C\_{2}^{-}Z\_{3}} + Z\_{2}\right) + \left(\alpha M\right)^{2}}\dot{U}\_{1\\_PP} \tag{42}$$

$$P\_{OUT\\_PP} = R\_L \cdot \left| \dot{I}\_6 \right|^2 = \left| \frac{R\_L \left( -\frac{M}{Z\_3 C\_2} \right)^2}{\left[ \left( Z\_1 + j\frac{1}{\alpha C\_1} \right) \left( \frac{1}{\alpha^2 C\_2} + Z\_2 \right) + (\alpha M)^2 \right]^2} \right|^2 \dot{U}\_{1\\_PP}^2 \tag{43}$$

$$\eta = \left| \frac{P\_{OUT\\_SP}}{S\_{M\_{-S}^\*,P^\*} \cos(\phi)} \right|$$

$$= \left| \frac{R\_L M^2}{Z\_3^2 C\_2^2 \left[ j\alpha C\_1 \left( Z\_1 + j \frac{1}{\alpha C\_1} \right) + \frac{\alpha^4 C\_2^2 Z\_3 M^2}{\alpha^2 C\_2^2 Z\_2 Z\_3 + 1} \right]^2 + \left( Z\_1 + j \frac{1}{\alpha C\_1} \right) + \frac{\alpha^4 C\_2^2 Z\_3 M^2}{\alpha^2 C\_2^2 Z\_2 Z\_3 + 1} \right| \cos(\phi) \Big|$$

Regarding of the system performance, it is similar to characteristics found for nonresonant magnetic coupling. Thus, this type of compensation is not suitable for applications where simultaneously high power with high efficiency is required to transmit power on large distances.

#### **7. Conclusion**

Based on the study, a table (**Table 1**) of key operating features for the discussed compensation topologies is composed. **Table 1** only covers the system behavior at chosen frequency and distance range while considering constant supply voltage and low value of loading resistance.

In **Table 1**, attribute "**A**" marks the ability to transfer high power to the load, attribute "**B**" means that the system operates with maximal efficiency at the same time when it transmits the maximal power. Attribute "**C**" distinguishes the compensation topology with higher fre‐ quency sensitivity and finally attribute "**D**" indicates the ability to reach theoretical trans‐ mitting efficiency, possibly higher than 90%.

**Table 1.** Overview of key performance features.

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= = æ öæ ö ç ÷ + ç ÷ + +

w

46 Wireless Power Transfer - Fundamentals and Technologies

.

w

( )

j

\_

*OUT SP*

w

.cos

\_

&

h

=

distances.

**7. Conclusion**

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w

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1 1

*L*

*R M*

1 2 2 2 1 23

2

Regarding of the system performance, it is similar to characteristics found for nonresonant magnetic coupling. Thus, this type of compensation is not suitable for applications where simultaneously high power with high efficiency is required to transmit power on large

Based on the study, a table (**Table 1**) of key operating features for the discussed compensation topologies is composed. **Table 1** only covers the system behavior at chosen frequency and distance range while considering constant supply voltage and low value of loading resistance.

In **Table 1**, attribute "**A**" marks the ability to transfer high power to the load, attribute "**B**" means that the system operates with maximal efficiency at the same time when it transmits

1 2 23 1 2 23

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*M*

1 1 ( ) *PP PP*

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& & & (41)

& & & (42)

2

1

2

2

( )

(44)

 j

cos

w

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& & (43)

1 2 23

w

According to **Table 1**, the noncompensated system and the system with parallel‒parallel compensation are both inappropriate for any type of wireless charger working at large dis‐ tance and higher level of transmitting energy. On the other hand, they achieve maximum efficiency precisely at the moment they deliver maximum power. Hence, they should be used for low-cost micropower battery chargers which moreover require no frequency tun‐ ing.

In contrast to other topologies, the series‒series compensation provides relatively high power even through a very large working air gap, but the efficiency reaches its peak only at offresonant frequencies. The system should be therefore operated at compromise between useable energy and the transmitting efficiency. Despite this fact, the system is suitable for highpower chargers for electric vehicles.

The series‒parallel compensation forms more disadvantages than advantages and therefore cannot be recommended for any battery charger. With respect to all evaluated categories, the parallel‒series compensation provides the best performance, but from power delivering point of view, this solution is much worse. However, it could be recommended for high-end frequency tuned micropower battery chargers.


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## Meet the editors

Syed Abdul Rehman Khan is a teacher of Supply Chain and Logistics Management. Dr. Khan achieved his Certified Supply Chain Professional (CSCP) certificate from the United States and successfully completed his PhD in China. Since 2018, Dr. Khan has been affiliated with Tsinghua University, Bejing, as a postdoctoral researcher. He has more than nine years of core experience in supply chain and logistics at industry and academ-

ic levels. He has attended several international conferences and has been invited as a keynote speaker in different countries. He has published more than 70 scientific research papers in well-renowned international journals and conferences. He is a regular contributor to conferences and workshops around the world. In addition, Dr. Khan has achieved two consecutive scientific innovation awards from the Education Department of Shaanxi Provincial Government, China. He holds memberships in the American Production and Inventory Control Society (APCIS); Production and Operations Management Society, India; Council of Supply Chain Management Professionals, USA; Supply Chain Association of Pakistan; and Global Supply Chain Council, China.

Ms. Zhang Yu is a doctoral candidate in Chang'an University, China. Her research interest is reverse logistics and packaging. She has published 15 research papers, including ISTP, SCIE, and EI indexed. Ms. Zhang also has attended several international conferences in China and Hong Kong.

Contents

**Section 1**

Death and Terror in 2020

Lake Chad Basin Region *by Cecilia Idika-Kalu*

to Violent Extremism *by Wilson Muna*

of Douala, Cameroon *by Nanche Billa Robert*

*by Simón Pedro Izcara Palacios*

**Section 2**

*by Syed Abdul Rehman Khan and Zhang Yu*

**Preface III**

Bioterrorism and Terrorism **1**

**Chapter 1 3**

**Chapter 2 9**

**Chapter 3 19**

Terrorism and Economic Losses **31**

**Chapter 4 33**

**Chapter 5 41**

**Chapter 6 57**

Introductory Chapter: The Outbreak of Coronavirus (COVID-19) -

The Socioeconomic Impact of the Boko Haram Insurgency in the

Terrorism in Emerging Economies: A Double-Edged Sword

The Survival Strategies of Poor Youth in the Metropolitan City

Mexican Migrant Smugglers and Foreign Terrorists

*by Syed Abdul Rehman Khan and Zhang Yu*

Family-Based Networks: Soft Policy Tools in Countering Radicalisation

## Contents

Preface

Terrorism is, in the broadest sense, the use of intentional violence for political purposes. It is primarily used to refer to violence during peacetime or in the context of war against non-combatants. The use of living organisms as a weapon has been seen throughout the history of humankind. But due to increasing trade wars between countries, the risk of biological terror attacks has become more real and more alarming. Bioterrorism is terrorism involving the intentional dissemination of biological agents. As in biological warfare, bioterrorism agents can be viruses, fungi, bacteria, toxins, insects or any other number of naturally occurring or

Before the start of Chinese New Year, the first patient with COVID-19 was diagnosed in Wuhan, China. Within a couple of days, the virus had spread all over China and infected more than 90,000 people. In instances such as this, it can be challenging for clinicians to discriminate between an act of terrorism and a naturally occurring outbreak. Therefore, this book discusses the effects of bioterrorism and terrorism on the socioeconomic sustainability of different countries around the

I would like to thank the contributors to this book for their hard work in crafting their valuable and innovative chapters. I would also like to thank IntechOpen and Author Service Manager Lada Bozic for their coordination and support in facilitat-

**Syed Abdul Rehman Khan, PhD**

School of Economics and Management,

Tsinghua University, Beijing, China

Chang'an University,

**Zhang Yu**

Xi'an, China

synthetic agents.

globe.

ing this project.


**7.2 Rotational movement**

Rotation occurs when the condyles rotate around a fixed point or axis during mandibular opening and closing. Rotational motion can occur in three reference planes: horizontal, vertical, and sagittal. Each of them occurs around a point called the axis [11].

vertical axis in the opposite condyle, which remains in said axis. This type of movement does not occur normally [16].

• Sagittal axis of rotation: Occurs when one of the condyles moves inferiorly while the other remains in the position of the terminal axis. This movement occurs in conjunction with other movements. Mathematical studies indicate that in this plane there is the same contact and muscle activity from one side to the other, so there are no alterations in dental occlusion that result in a joint without load [11, 16].

The amount of condylar rotation does not differ between men and women. A finding that contrasts with the greater maximum interincisal opening of men compared to women due to differences in jaw length. In fact, with the same degree of rotation, the greater the length of the mandible, the greater the opening of the mouth. Consequently, the degree of interincisal opening cannot be considered as a measure of joint mobility or laxity, unless corrected for mandibular size [8].

### **7.3 Translational motion**

Translation can be defined as a movement in which every point of the object t simultaneously has the same speed and direction. In the masticatory system, it occurs when the mandible protrudes. During normal movements, rotation and translation occur simultaneously, as the mandible rotates in one or more axes, each of the axes is changing orientation in space [16].

The total movement of the mandible does not consist only of rotation and translation. Side-to-side or eccentric bodily movement of the mandible and rotation and translation of the joints indicate that the mandible acts as a free-moving or floating; structure. Controlled by pairs of complementary and opposing functional muscle groups that gradually exert impulse force with numerous force vectors, the threedimensional movement of the mandible with a dual-operation joint system is unlike any other orthopedic system in the body [17].

Classical records analyzed mandibular movements in terms of their geometry, using mechanical systems. Posselt designed an instrument called a gnatho-tensiometer, which could record border movements in all three planes, obtaining the Posselt diagram. Currently, technology has made it possible to improve position tracking techniques and thus be able to analyze mandibular kinematics with high spatial and temporal resolution (**Figure 1**) [18].

#### **7.4 Temporomandibular joint and its relationship with occlusion**

Movement is not only guided by the shape of the bones, muscles, and ligaments, but also by the occlusion of the teeth [1]. The Glossary of Prosthodontic Terms defines occlusion as the static relationship between the chewing surfaces of the maxillary and mandibular teeth. Dental contact has to be studied from a functional perspective and a more adequate definition of occlusion would be the biological and dynamic relationship of the components of the masticatory system that determines dental relationships [19].

Occlusion comprises a wide range of topics, the biomechanics of occlusal contact between two teeth with different cusp inclinations form a complex system [16]. From a clinical point of view, TMJ changes including intracapsular exudate and joint tissue

#### **Figure 1.** *Posselt diagram.*

loss can result in occlusal changes such as anterior or posterior open bites. It is important to mention that a particular occlusal scheme is not a determinant of disease. There is no evidence to suggest that one scheme predominates over another. Group functions compared to canine guides cause less condylar displacement, this displacement is small and has no clinical significance [19].

The range of vertical movement is dictated by anterior determinants such as overbite and posterior determinants such as TMJ condylar guidance. From a biomechanical point of view, anterior versus posterior determinants have a greater influence on tooth contact due to their proximity to the teeth. On the other hand, the condylar guide will influence when the molars are in contact or close to contact during mandibular movements [19].


3.1.4. Packet Modification

The key part of fast network recovery technology is the fast detection of the failure and the subsequent means of its notification to the other routers which were affected by the failure (disrupted routing). In some FRR mechanisms, specific link failure information is distributed by the following techniques:

It should be noted that packet modification causes various compatibility problems as well as problems with exceeding the Maximum Transmission Unit (MTU) on some network links [26].


**1. Introduction**

Toxocariosis is a neglected zoonotic disease transmitted from dogs and cats to humans. This is mainly caused by the presence and action of the nematode *Toxocara canis* (*T. canis*) and less frequently by *Toxocara cati* (*T. cati* syn. *T. mystax*). *T. canis* uses canines, mainly puppies, as its definitive host, and *T. cati* uses kittens. In addition, they use a wide variety of paratenic hosts, including pigs, sheep, rabbits, rats, mice, other mammals, chickens, and other birds. In humans, the infection is accidental, and the parasite behaves similarly as it does in paratenic hosts. Some invertebrates, such as earthworms and cockroaches, can also have *Toxocara* larvae in their tissues or their gut [1, 2].

Adult *T. canis* worms live in the small intestine of puppies. The females measure from 10 to 18 cm, and the males measured from 4 to 10 cm. At the anterior end, they have three small lips that do not protrude beyond the diameter of the body.

A dentigerous border can be seen on the inner surface of each lip.  The posterior end in males ends coiled toward the ventral part and has a terminal narrowing in the form of an appendix, and a pair of small and symmetrical spicules (0.75–0.95 mm) are observed. In females, the vulva is located approximately in the middle of the body, and the posterior end ends at a straight, blunt point. *T. cati* adults are very similar to *T. canis*; the cervical fins are broader and convex, giving the anterior end a more marked arrowheaded appearance; males are 3–6 cm long, and females are 4–10 cm long [3].


3.3.1. Selection Time

In the process of the experimental data analysis, we set the target level (3 × 3, 4 × 4, 5 × 5, 6 × 6, 7 × 7, and 8 × 8) and the target region (A, B, C, and D) as independent variables. In this way, we performed repeated measurements ANOVAs (α = 0.05) on the time and accuracy of the target selection. The target selection time was defined as beginning from when the user clicked the Start button or pressed the left Ctrl button to when the user pressed the left Ctrl button again.

There was a main effect on the average time of the different regions (F2.058, 22.634 = 11.460, *p* < 0.001), see Figure 5a. The post hoc tests showed that there were no significant differences among the regions (*p* > 0.05) except for between regions B and C (*p* = 0.035) and regions B and D (*p* < 0.001). Region B had the fastest completion time, and region D had the slowest completion time.

**Figure 5.** The average time of Experiment 1. The error bars represent a 95% confidence interval: (**a**) average selection time with different regions; (**b**) average selection time with different target size levels; and (**c**) average selection time for different target size levels and different regions (A, B, C, and D).

There was a main effect for the average time of the different levels of target size (F2.321, 25.531 = 20.714, *p* < 0.001), see Figure 5b. The post hoc tests showed that the shortest time was for the 3 × 3 level, and the longest was for the 7 × 7 level. There were no significant differences between the 3 × 3, 4 × 4, and 5 × 5 levels (*p* > 0.119). There were no significant differences between the 6 × 6, 7 × 7, and 8 × 8 levels (*p* > 0.05).

Further analysis of the level of target size × target region on selection time showed there was no significant interaction (F4.272, 46.992 = 1.601, *p* = 0.187), see Figure 5c. When the users selected a target in the 3 × 3 level, the shortest selection time was needed on average, while the 7 × 7 level had the longest time.

#### 3.3.2. Selection Error Rate

The percentage of trials in which subjects made erroneous selections was defined as the selection error rate.

As shown in Figure 6a, there was a main effect on the average error rate of the different regions (F3, 33 = 4.240, *p* = 0.012). Post hoc tests showed no significant differences among all the regions (*p* > 0.052), except between regions B and D (*p* = 0.033). Region D had the lowest completion error rate, and region C had the highest completion error rate.

**Figure 6.** The error rate of Experiment 1. The error bars represent a 95% confidence interval: (**a**) average error rate with different regions; (**b**) average error rate with different target size levels; and (**c**) average error rate for different target size levels under different regions (A, B, C, and D).

As shown in Figure 6b, different target sizes had no significant effect on the average error rate (F2.402,26.419 = 2.617, *p* = 0.083). The post hoc tests showed no significant differences among all the target size levels (*p* > 0.157). The 3 × 3 level had the lowest error rate, and the 8 × 8 level had the highest. The higher the target size level, the higher the error rate when selecting the target. The largest increase in the error rate for adjacent levels was from the 5 × 5 to the 6 × 6 level. Thus, a target size of 5 × 5 (60.8 mm length and 48 mm width) provided a threshold for the most selections without a noticeable change in error rate.

Further analysis of what effect target size level × target region had on selection error rate showed there was no significant interaction (F4.256, 46.816 = 1.665, *p* = 0.171), see Figure 6c. The 3 × 3 target size level had the lowest selection error rate. The second-lowest selection error rate was the 4 × 4 level. The 8 × 8 level produced the highest error rate. The participants had the lowest error rate (0%) when the target region was D (bottom-right corner) and the highest error rate (3.54%) when the target region was C (bottom-left corner).

In previous literature [19], the author studied pointing at virtual buttons. The space is divided into 5 different sizes according to the angle, that is, the number of buttons. The experimental results show that the error rates are 0, 3.6%, 2.2%, 16.0%, 3.2%, respectively. As shown in Figure 6a, the error rates of our results are 0.31%, 0.83%, 1.39%, 2.78%, 2.22%, 3.06%, and the overall error rate is better. We also divided the regions, discussed the situation of each region, and the comprehensive situation of region and size. The literature only considers the error rate and not the task completion time. We comprehensively analyze the error rate and time and give suggestions for designing interactive technologies based on spatial regions, which are more convincing. Next, we studied the division of spatial regions in the absence of vision and give suggestions for designing interactive technologies based on spatial regions in the absence of vision.

#### *3.4. Comparative Experiment 1*

The participants in Experiment 1 were the same as in the pilot study, so they were trained and familiar with the experiment. To eliminate this influence, we invited 12 external participants who didn't know the experiment in advance. The experiment process was the same as experiment 1.

#### 3.4.1. Selection Time

There was a main effect on the average time of the different regions (F2.058, 22.634 = 11.168, *p* < 0.001), see Figure 7a. The post hoc tests showed that there were no significant differences among the regions (*p* > 0.05) except for between regions B and C (*p* = 0.023) and regions B and D (*p* = 0.002). Region B had the fastest completion time, and region D had the slowest completion time.

**Figure 7.** The average time of comparative Experiment 1. The error bars represent a 95% confidence interval: (**a**) average selection time with different regions; (**b**) average selection time with different target size levels; and (**c**) average selection time for different target size levels and different regions (A, B, C, and D).

There was a primary effect regarding the average time of the different levels of target size (F2.321, 25.531 = 20.870, *p* < 0.001), see Figure 7b. The post hoc tests showed that the shortest time was for the 3 × 3 level, and the longest was for the 7 × 7 level. There were no significant differences among the 3 × 3, 4 × 4, and 5 × 5 levels (*p* > 0.37). There were no significant differences among the 6 × 6, 7 × 7, and 8 × 8 levels (*p* > 0.05).

Further analysis of the level of target size × target region on selection time showed there was no significant interaction (F4.272, 46.992 = 1.635, *p* = 0.178), see Figure 7c. When the users selected a target in the 3 × 3 level, the shortest selection time was needed on average, while the 7 × 7 level had the longest time.

## 3.4.2. Selection Error Rate

As shown in Figure 8a, there was a primary effect on the average error rate of the different regions (F3, 33 = 3.996, *p* = 0.016). Post hoc tests showed no significant differences among all the regions (*p* > 0.063). Region D had the lowest completion error rate, and region C had the highest completion error rate.

As shown in Figure 8b, there was no significant effect for the average error rate of the different target sizes (F2.109, 23.199 = 2.184, *p* = 0.133). The post hoc tests showed no significant differences among all the target size levels (*p* > 0.122). The 3 × 3 level had the lowest error rate, and the 8 × 8 level had the highest.

Further analysis of target size level × target region on selection error rate showed there was no significant interaction (F4.9, 53.903 = 1.628, *p* = 0.17), see Figure 8c.

**Figure 8.** The error rate of comparative Experiment 1. The error bars represent a 95% confidence interval: (**a**) average error rate with different regions; (**b**) average error rate with different target size levels; and (**c**) average error rate for different target size levels under different regions (A, B, C, and D).

Compared with Experiment 1, the error rate of this experiment was slightly higher, and the average time was slightly longer, caused by the fact that new participants were not familiar with the experiment. The results showed that the regions (and levels) with the highest or lowest error rates were the same as Experiment 1. The regions (and levels) with the fastest or slowest average time were the same as Experiment 1, as shown in Figures 7 and 8.

#### **4. Experiment 2: Non-Visual Scenario**

To test the accuracy when the users performed the task with an eyes-free scenario, we set voice guidance for a visually impaired individual. The position of the users' hand mapping to the current block was shown on the experiment screen in real time, and the target block turned from green to red while the current block overlapped it.

## *4.1. Participants & Apparatus*

The participants and apparatus in Experiment 2 were the same as in the pilot study.

#### *4.2. Task & Procedure*

The design and tasks were almost the same as in Experiment 1. The difference in Experiment 2 was that there was no visual feedback for the users, only voice guidance. The beginning guide audio was "The target block is X", and it would then announce the number of the block of the users' hand in real time. The participants already knew the number of the target block. This cycle would continue until the task was completed, and then the audio would announce, "This round of the experiment ends". Before the formal experiment, participants were allowed to warm up with a practice session until they could understand and perform the task correctly. In total, the experiment consisted of the following: 12 subjects × 6 target size levels × 4 target regions × 2 blocks × 3 repetitions = 1728 target selection trials.

#### *4.3. Results*

#### 4.3.1. Selection Time

We found a main effect on the average time of different regions (F3, 33 = 13.496, *p* < 0.001), see Figure 9a. The post hoc tests showed a significant difference between regions A and C (*p* = 0.007) and regions A and D (*p* = 0.002). There was a significant difference between regions B and C (*p* = 0.042) and regions B and D (*p* = 0.002). Other regions had no significant differences (*p* > 0.975). Region B had the fastest completion time, and region D had the slowest.

**Figure 9.** Average times of Experiment 2. Error bars represent a 95% confidence interval: (**a**) average selection time with different regions; (**b**) average selection time with different target size levels; and (**c**) average selection time for different target size levels under different regions (A, B, C, and D).

As shown in Figure 9b, there was a primary effect on the average time of the different target sizes (F2.222, 24.445 = 24.893, *p* < 0.001). A post hoc test showed no significant difference between the 3 × 3 and 4 × 4 levels (*p* = 0.231). There was no significant difference between the 5 × 5 and 6 × 6 levels (*p* = 0.270). There was no significant difference between the 7 × 7 and 8 × 8 levels (*p* = 0.142). The 3 × 3 level had the fastest completion time, and the 8 × 8 level had the slowest completion time. The higher the target size level, the longer the time needed to select the target. The largest increase in selection time for adjacent levels was from the 6 × 6 to the 7 × 7 level. Thus, the target size of 6 × 6 (50.67 mm length and 40 mm width) provided a threshold of the most selections without a noticeable change in selection time.

Further analysis of target size level × target region on selection time showed there was no significant interaction (F4.912, 54.027 = 1.575, *p* = 0.184), see Figure 9c. The shortest selection time was needed when users selected a target in the 3 × 3 level. The second shortest level was the 4 × 4 level, while the 8 × 8 level took the longest time.

#### 4.3.2. Selection Error Rate

There was no significant effect concerning the average error rate of the different regions (F3, 33 = 0.909, *p* = 0.447), see Figure 10a. The post hoc tests showed no significant differences among all the regions (*p* = 0.670). Region B had the lowest completion error rate, and region C had the highest.

**Figure 10.** The error rate of Experiment 2. The error bars represent a 95% confidence interval: (**a**) average error rate with different regions; (**b**) average error rate with different target size levels; (**c**) average error rate for different target size levels under different regions (A, B, C, and D).

As shown in Figure 10b, there was no significant effect concerning the average error rate of the different target sizes (F5, 55 = 4.388, *p* = 0.002). The post hoc tests showed no significant differences among all the levels of target size (*p* > 0.156), except for the 3 × 3 level and 6 × 6 level (*p* = 0.030). The 3 × 3 level had the lowest completion error rate, and the 8 × 8 level had the highest completion error rate. The higher the target size

level, the higher the error rate needed to select a target. The largest increase in the selection error rate for adjacent levels was from the 4 × 4 to the 5 × 5. Thus, a target size of 4 × 4 (76 mm length and 60 mm width) provided a threshold of the most selections without a noticeable change in selecting error rate.

Further analysis of the target size level × target region on selection error rate showed there were no significant interaction (F5.122, 56.337 = 1.524, *p* = 0.196), see Figure 10c. On average, the 3 × 3 target size level had the lowest selection error rate. The second-lowest selection error rate was for the 4 × 4 level. The 8 × 8 level produced the highest error rate. The participants reached the lowest selection error rate (3.54%) when the target region was B (upper-right corner), and the highest error rate (6.31%) when the target region was C (bottom-left corner).

#### *4.4. Comparative Experiment 2*

The participants in Experiment 2 were the same as in the pilot study, so they were trained and familiar with the experiment. To eliminate this influence, we invited 12 external participants who did not know the experiment in advance. The experiment process was the same as experiment 2.

#### 4.4.1. Selection Time

We found a main effect on the average time of different regions (F3, 33 = 13.474, *p* < 0.001), see Figure 11a. The post hoc tests showed a significant difference between regions A and C (*p* = 0.007) and regions A and D (*p* = 0.003). There was a significant difference between regions B and C (*p* = 0.042) and regions B and D (*p* = 0.002). Other regions had no significant differences among them (*p* > 0.994). Region B had the fastest completion time, and region D had the slowest completion time.

**Figure 11.** The average time of comparative Experiment 2. The error bars represent a 95% confidence interval: (**a**) average selection time with different regions; (**b**) average selection time with different target size levels; and (**c**) average selection time for different target size levels and different regions (A, B, C, and D).

As shown in Figure 11b, there was a main effect on the average time of the different target sizes (F2.222, 24.445 = 25.009, *p* < 0.001). A post hoc test showed no significant difference between the 3 × 3 and 4 × 4 levels (*p* = 0.227). There was no significant difference between the 5 × 5 and 6 × 6 levels (*p* > 0.05). There was no significant difference between the 7 × 7 and 8 × 8 levels (*p* > 0.05). The 3 × 3 level had the fastest completion time, and the 8 × 8 level had the slowest completion time.

Further analysis of target size level × target region on selection time showed there was no significant interaction (F4.912, 54.027 = 1.574, *p* = 0.184), see Figure 11c.

#### 4.4.2. Selection Error Rate

There was no significant effect for the average error rate of the different regions (F3, 33 = 0.883 *p* = 0.46), see Figure 12a. The post hoc tests showed no significant differences among all the regions (*p* > 0.05). Region B had the lowest completion error rate, and region C had the highest completion error rate.

**Figure 12.** The error rate of comparative Experiment 1. The error bars represent a 95% confidence interval: (**a**) average error rate with different regions; (**b**) average error rate with different target size levels; and (**c**) average error rate for different target size levels under different regions (A, B, C, and D).

As shown in Figure 12b, there was no significant effect for the average error rate of the different target sizes (F5, 55 = 3.193, *p* = 0.013). The post hoc tests showed no significant differences among all the levels of target size (*p* > 0.109). The 3 × 3 level had the lowest completion error rate, and the 8 × 8 level had the highest completion error rate.

Further analysis of the target size level × target region on selection error rate showed there were no significant interaction (F6.461,71.069 = 1.085, *p* = 0.381), see Figure 12c. Compared to Experiment 2, the error rate of this experiment was slightly higher, and the average time was slightly longer, caused by the fact that new participants were not familiar with the experiment. The results showed that the regions (and levels) with the highest or lowest error rates were the same as in Experiment 2. The regions (and levels) with the fastest or slowest average time were the same as Experiment 2, as shown in Figures 11 and 12.

#### **5. Discussion & Conclusions**

In this work, we analyzed the users' common operational area regarding partitioning and the difference in spatial controllability between a sighted and a visually impaired individual. We introduced three experiments and a pilot study concerning the common spatial range and the thresholds of the target size level in the spatial region for a sighted and visually impaired individual. We compared the speed and accuracy of the target dimensions of six different levels and the difference in speed and accuracy among the four azimuth regions of A, B, C, and D in both visual and non-visual scenarios. Many of our performance study results were statistically significant, which allows us to draw many meaningful conclusions about human-computer interaction in spatial regions that can be used for designing techniques for sighted and visually impaired individuals. This paper focused on systematically analyzing the common operational range of one dimension and the threshold of two dimensions. The results are as follows:

Based on the above results and findings, we have developed a set of preliminary guidelines regarding target selection in spatial partitioning scenarios:

In the case of vision, researchers can refer to the design suggestions in Table 3 when studying the space operation capabilities between users and computer screens or designing interactive technologies based on spatial regions. In the case of non-vision, researchers can refer to the design suggestions in Table 4 when studying the space operation capabilities between users and computer screens or designing interactive technologies based on spatial regions.

**Table 3.** Design suggestions for selecting spatial targets under visual conditions.

**Table 4.** Design suggestions for selecting spatial targets under non-visual conditions.

Based on these results, we propose two techniques for two different application scenarios, described in the following paragraphs.

A spatial gesture recognition technique for surgery can help users select targets by using spatial region cognition and hand gestures during surgery. This technique is designed based on the partitioning strategies of a common operational spatial region array. This technique can meet the strict requirements of sanitary conditions during surgery (as opposed to a touchscreen and most other existing interfaces).

Non-visual selection is a system integrated with screen reading software allowing a visually impaired person to select targets easily. This technique is designed based on the partitioning strategies of a common operational spatial region array. Users can use this system to interact with the internet and web more easily. In addition, the user no longer needs a keyboard because this system uses Leap Motion to detect a users' hand motions and provides voice guidance when choosing targets and to do further work.

In the future, we will further expand the results of this study and contribute to technology accessibility for visually impaired individuals, including the exploration of a threshold for three-dimensional interaction.

**Author Contributions:** Methodology, validation, data curation, writing—review and editing, visualization, H.W.; conceptualization, formal analysis, resources, supervision, project administration, funding acquisition, J.Y.; software, data curation, methodology, writing—original draft preparation, Y.Z.; formal analysis, resources, supervision, project administration, investigation, Y.H.; investigation, supervision, methodology, resources, X.Z.; methodology, resources, formal analysis, supervision, S.W. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was funded by the National Natural Science Fund, grant (61741206).

**Acknowledgments:** The author thanks the editor and others for their comments and suggestions for this article.

**Conflicts of Interest:** The authors declare no conflict of interest.

## **References**