**2.1 Series-series compensation**

The system using resonant coupling fully compensates for the scattering fields of

Energy transmission through capacitive coupling is currently used relatively little due to limitations on the transmission distance, which is limited by the level of tenths of a millimeter. This method is mainly used for charging consumer electronics such as tablets, laptops and more. They also have great potential in the field of medicine for charging various implants. Capacitive coupling is a phenomenon occurring between all conductive objects, i.e. between systems between which there is a mutual difference of potentials and between them there is an environment with

This work aims to point out the main design issues related to wireless power transmission and demonstrate their operational characteristics. An important aspect in this area is undoubtedly interaction of living organisms with a strong electromagnetic field, and therefore it is necessary to pay attention also legislation and hygiene standards [12–15]. Another goal is to provide a clear mathematical description of the system using intuitive methods for circuit analysis. Mathematical models must consider, in addition to the coupling itself, the inverters (inverter and rectifier) on the primary and secondary side of the system. An equally important goal of the work is experimental verification of all achieved theoretical conclusions. For this purpose, it was necessary to develop a prototype of a WPT charging system capable to supply sufficient power needed to charge conventional electric car. The text is supplemented by accompanying graphics that illustrates efficiency characteristics and also analysis of the spatial distribution of the electromagnetic field at different states of the system.

There are four basic configurations of the primary and secondary side of the WPT system to realize resonant compensation of the leakage inductance [16–18]. This chapter focuses on investigating the properties of possible compensation methods, which include serial-serial, serial-parallel, parallel-serial, parallel–parallel.

*R*<sup>Z</sup> Series compensation: 5÷200 Ω Parallel comp.:

5 Ω÷2.5 kΩ

the coupling coils, thus significantly extending the working distance while maintaining high energy transfer efficiency. Thanks to its advantages, resonant coupling is used mainly in the field of electromobility, where it allows charging with high power and in the case of a variable load, it can be easily frequency-adjusted for

*Wireless Power Transfer – Recent Development, Applications and New Perspectives*

optimal efficiency [7, 8].

a positive dielectric constant (permittivity) [9–11].

**2. Basic resonant coupling techniques**

**Parameter Value** *R*<sup>1</sup> 0.45 Ω *R*<sup>2</sup> 0.45 Ω *L*<sup>1</sup> 145.6 μH *L*<sup>2</sup> 145.6 μH *C*<sup>1</sup> 3.1 nF *C*<sup>2</sup> 3.1 nF *k* 0÷0.1

*Ub* 100 V

*Circuit parameters for the evaluation of compensation techniques.*

**Table 1.**

**44**

Serial-series compensation uses two external capacitors *C1* and *C2* connected in series with the primary and secondary windings. The circuit model of the system is shown in **Figure 1**.

The system is powered by an inverter with a rectangular voltage profile with amplitude um1, and therefore the circuit must be described by a system of integrodifferential equations forming a full-fledged dynamic model.

$$\begin{aligned} -u\_1 + \frac{1}{C\_1} \int\_0^t i\_1 dt + u\_{C\_1(0)} + R\_1 i\_1 + L\_1 \frac{di\_1}{dt} + M \frac{di\_2}{dt} &= 0 \\ -L\_2 \frac{di\_2}{dt} - M \frac{di\_1}{dt} - R\_2 i\_2 - \frac{1}{C\_2} \int\_0^t i\_2 dt - u\_{C\_2(0)} - R\_2 i\_2 &= 0 \end{aligned} \tag{1}$$

All models will be derived for the fundamental harmonic and therefore we can use Eq. (1) to describe the model, while the inverter voltage can be considered in the form of (2).

$$
\dot{U}\_1 = \frac{2\sqrt{2}}{\pi} u\_{m1} \tag{2}
$$

The solution we get loop currents, from which it is possible to further determine all operating variables of the system (3).

$$
\begin{bmatrix} \dot{I}\_{\rm S1} \\ \dot{I}\_{\rm S2} \end{bmatrix} = \begin{bmatrix} R\_1 + j \left( a\omega L\_1 - \frac{1}{a\omega C\_1} \right) & -j\omega \mathbf{M} \\\\ -j\alpha \mathbf{M} & R\_2 + R\_Z + j \left( a\omega L\_2 - \frac{1}{a\omega C\_2} \right) \end{bmatrix} \begin{bmatrix} \dot{U}\_1 \\ \mathbf{0} \end{bmatrix} \tag{3}
$$

For a better idea, we draw the efficiency and power on the load depending on the frequency and the coupling factor, respectively the load. This creates two pairs of maps in which two functions are plotted separately:

#### **Figure 1.**

*Simplified equivalent circuit for series–series compensation, left – Circuit with initial variables, right – Circuit suited for loop current analysis.*

*Wireless Power Transfer – Recent Development, Applications and New Perspectives*

$$P\_2 = f\left(\left.f\_{sw},k\right), P\_2 = f\left(\left.f\_{sw}, R\_Z\right), \text{and } \eta = f\left(\left.f\_{sw},k\right), \eta = f\left(\left.f\_{sw}, R\_Z\right)\right.\right.\tag{4}$$

The first map will consider a constant load, which will be set as optimal for the working distance corresponding to *k* = 0.1 [19]. The optimal load for the map was determined based on the relationship:

$$R\_{Zs-\text{efficiency}} = \sqrt{\frac{R\_2 \left(R\_1 R\_2 + M^2 \alpha^2\right)}{R\_1}} \doteq 22\Omega. \tag{5}$$

�*u*<sup>1</sup> þ

*DOI: http://dx.doi.org/10.5772/intechopen.95749*

�*L*<sup>2</sup> *di*<sup>2</sup> *dt* � *<sup>M</sup> di*<sup>1</sup>

� 1 *C*2 ð*t* 0

*<sup>R</sup>*<sup>1</sup> <sup>þ</sup> *<sup>j</sup> <sup>ω</sup>L*<sup>1</sup> � <sup>1</sup>

*ωC*<sup>1</sup>

� �

The optimal load is then given by the Eq. (8)

*RZp*�*efficiency* <sup>¼</sup> *<sup>L</sup>*2*<sup>ω</sup>*

**2.3 Parallel-series compensation**

unknowns for the description (9).

´*IS*1 ´*IS*2 ´*IS*3

**Figure 5**.

**Figure 4.**

**47**

*S-P compensation.*

2 6 4

1 *C*1 ð*t* 0

*Theoretical and Practical Design Approach of Wireless Power Systems*

*i*1*dt* þ *uC*1ð Þ <sup>0</sup> þ *R*1*i*<sup>1</sup> þ *L*<sup>1</sup>

ð Þ *i*<sup>3</sup> � *i*<sup>2</sup> *dt* � *uC*2ð Þ <sup>0</sup> � *RZi*<sup>3</sup> ¼ 0

�*jωM R*<sup>2</sup> <sup>þ</sup> *<sup>j</sup> <sup>ω</sup>L*<sup>2</sup> � <sup>1</sup>

*R*1*R*<sup>2</sup>

The resulting waveforms are graphically summarized in **Figure 4**.

<sup>0</sup> *<sup>j</sup>*

1 *C*2 ð*t* 0

For the calculation by the loop current method, the equations take the form (7).

*ωC*<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q � � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>R</sup>*<sup>2</sup> *<sup>R</sup>*1*R*<sup>2</sup> <sup>þ</sup> *<sup>M</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup> *<sup>L</sup>*<sup>2</sup>

Parallel–series compensation is practically only like the previous variant, which uses two resonant capacitors connected in parallel with the coupling coil on the primary side and in series on the secondary side. The circuit model is apparent from

As in the previous case, it is enough to compile three equations of three

*Dependency of output power (left) and system efficiency (right) on frequency and power transfer distance for*

q � �

*dt* � *<sup>R</sup>*2*i*<sup>2</sup> <sup>þ</sup>

*di*<sup>1</sup> *dt* <sup>þ</sup> *<sup>M</sup> di*<sup>2</sup>

�*jωM* 0

*ωC*<sup>2</sup> � � *j*

<sup>2</sup>*R*<sup>1</sup> <sup>þ</sup> *<sup>M</sup>*<sup>2</sup>

*ω*2

ð Þ *i*<sup>3</sup> � *i*<sup>2</sup> *dt* þ *uC*2ð Þ <sup>0</sup> ¼ 0

*dt* <sup>¼</sup> <sup>0</sup>

*ωC*<sup>2</sup>

þ *RZ*

2 6 4

≐2176*Ω:* (8)

� � � � �

�*j ωC*<sup>2</sup>

*R*2

(6)

The resulting maps are shown in **Figure 2**.

#### **2.2 Series-parallel compensation**

Serial-parallel compensation uses two resonant capacitors connected in series with a coupling coil on the primary side and in parallel on the secondary side (**Figure 3**). In the left part you can see the diagram for the dynamic model and in the right part its simplification for the supply of the harmonic course of the voltage.

Unlike the previous circuit, it is now necessary to compile three equations of three unknowns. The integrodifferential form is given in the (6).

#### **Figure 2.**

*Dependency of output power (left) and system efficiency (right) on frequency and power transfer distance for S-S compensation.*

#### **Figure 3.**

*Simplified equivalent circuit for series–parallel compensation, left – Circuit with initial variables, right – circuit suited for loop current analysis.*

*Theoretical and Practical Design Approach of Wireless Power Systems DOI: http://dx.doi.org/10.5772/intechopen.95749*

*<sup>P</sup>*<sup>2</sup> <sup>¼</sup> *f fsw*, *<sup>k</sup>* � �, *<sup>P</sup>*<sup>2</sup> <sup>¼</sup> *f fsw*, *RZ*

determined based on the relationship:

**2.2 Series-parallel compensation**

**Figure 2.**

**Figure 3.**

**46**

*right – circuit suited for loop current analysis.*

*S-S compensation.*

*RZs*�*efficiency* ¼

three unknowns. The integrodifferential form is given in the (6).

The resulting maps are shown in **Figure 2**.

� �, and *<sup>η</sup>* <sup>¼</sup> *f fsw*, *<sup>k</sup>* � �, *<sup>η</sup>* <sup>¼</sup> *f fsw*, *RZ*

The first map will consider a constant load, which will be set as optimal for the working distance corresponding to *k* = 0.1 [19]. The optimal load for the map was

Serial-parallel compensation uses two resonant capacitors connected in series with a coupling coil on the primary side and in parallel on the secondary side (**Figure 3**). In the left part you can see the diagram for the dynamic model and in the right part its simplification for the supply of the harmonic course of the voltage. Unlike the previous circuit, it is now necessary to compile three equations of

*Dependency of output power (left) and system efficiency (right) on frequency and power transfer distance for*

*Simplified equivalent circuit for series–parallel compensation, left – Circuit with initial variables,*

s

*Wireless Power Transfer – Recent Development, Applications and New Perspectives*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>R</sup>*<sup>2</sup> *<sup>R</sup>*1*R*<sup>2</sup> <sup>þ</sup> *<sup>M</sup>*<sup>2</sup> *ω*<sup>2</sup> � � *R*1

� � (4)

≐ 22*Ω:* (5)

$$-u\_1 + \frac{1}{C\_1} \int\_0^t i\_1 dt + u\_{C\_1(0)} + R\_1 i\_1 + L\_1 \frac{di\_1}{dt} + M \frac{di\_2}{dt} = \mathbf{0}$$

$$-L\_2 \frac{di\_2}{dt} - M \frac{di\_1}{dt} - R\_2 i\_2 + \frac{1}{C\_2} \int\_0^t (i\_3 - i\_2) dt + u\_{C\_2(0)} = \mathbf{0} \tag{6}$$

$$-\frac{1}{C\_2} \int\_0^t (i\_3 - i\_2) dt - u\_{C\_2(0)} - R\_2 i\_3 = \mathbf{0}$$

For the calculation by the loop current method, the equations take the form (7).

$$
\begin{bmatrix}
\dot{I}\_{S1} \\
\dot{I}\_{S2} \\
\dot{I}\_{S3}
\end{bmatrix} = \begin{bmatrix}
R\_1 + j\left(aL\_1 - \frac{1}{a\epsilon \mathbf{C}\_1}\right) & -j\alpha\mathbf{M} & \mathbf{0} \\\\ -j\alpha\mathbf{M} & R\_2 + j\left(aL\_2 - \frac{1}{a\epsilon \mathbf{C}\_2}\right) & \frac{j}{a\epsilon \mathbf{C}\_2} \\\\ \mathbf{0} & \frac{j}{a\epsilon \mathbf{C}\_2} & \frac{-j}{a\epsilon \mathbf{C}\_2} + R\_2
\end{bmatrix} \begin{bmatrix}
\dot{U}\_1 \\
\mathbf{0} \\
\mathbf{0}
\end{bmatrix} \tag{7}
$$

The optimal load is then given by the Eq. (8)

$$R\_{Zp-\text{efficancy}} = \frac{L\_2 \alpha \sqrt{R\_1 R\_2^2 + \alpha^2 \left(L\_2^2 R\_1 + M^2 R\_2\right)}}{\sqrt{R\_2 \left(R\_1 R\_2 + M^2 \alpha^2\right)}} \doteq 2176 \Omega. \tag{8}$$

The resulting waveforms are graphically summarized in **Figure 4**.

#### **2.3 Parallel-series compensation**

Parallel–series compensation is practically only like the previous variant, which uses two resonant capacitors connected in parallel with the coupling coil on the primary side and in series on the secondary side. The circuit model is apparent from **Figure 5**.

As in the previous case, it is enough to compile three equations of three unknowns for the description (9).

**Figure 4.**

*Dependency of output power (left) and system efficiency (right) on frequency and power transfer distance for S-P compensation.*

#### **Figure 5.**

*Simplified equivalent circuit for parallel - series compensation, left – Circuit with initial variables, right – Circuit suited for loop current analysis.*

$$\begin{aligned} -u\_1 + \frac{1}{C\_1} \int\_0^t (i\_1 - i\_2) dt + u\_{C\_1(0)} &= 0\\ -\frac{1}{C\_1} \int\_0^t (i\_1 - i\_2) dt - u\_{C\_1(0)} + R\_1 i\_2 + L\_1 \frac{di\_2}{dt} + M \frac{di\_3}{dt} &= 0\\ -L\_2 \frac{di\_3}{dt} - M \frac{di\_2}{dt} - R\_2 i\_3 - R\_2 i\_3 &= 0 \end{aligned} \tag{9}$$

**2.4 Parallel-parallel compensation**

*right – Circuit suited for loop current analysis.*

�*u*<sup>1</sup> þ

� 1 *C*1 ð*t* 0

�*L*<sup>2</sup> *di*<sup>3</sup> *dt* � *<sup>M</sup> di*<sup>2</sup>

� 1 *C*2 ð*t* 0

�*j ωC*<sup>1</sup>

**2.5 Overall comparison**

topologies and is not regulated.

*j ωC*<sup>1</sup>

´*IS*1 ´*IS*2 ´*IS*3 ´*IS*4

**Figure 7.**

**49**

1 *C*1 ð*t* 0

Parallel–parallel compensation uses two resonant capacitors connected in parallel to both coupling coils. The circuit model can be seen in **Figure 7**.

*Simplified equivalent circuit for parallel - parallel compensation, left – Circuit with initial variables,*

*Theoretical and Practical Design Approach of Wireless Power Systems*

*DOI: http://dx.doi.org/10.5772/intechopen.95749*

ð Þ *i*<sup>1</sup> � *i*<sup>2</sup> *dt* þ *uC*1ð Þ <sup>0</sup> ¼ 0

ð Þ *i*<sup>2</sup> � *i*<sup>1</sup> *dt* � *uC*1ð Þ <sup>0</sup> þ *R*1*i*<sup>2</sup> þ *L*<sup>1</sup>

ð Þ *i*<sup>4</sup> � *i*<sup>3</sup> *dt* � *uC*2ð Þ <sup>0</sup> � *RZi*<sup>4</sup> ¼ 0

After stabilization we can rewrite the equations into the form (12).

*ωC*<sup>1</sup>

<sup>0</sup> �*jωM R*<sup>2</sup> <sup>þ</sup> *<sup>j</sup> <sup>ω</sup>L*<sup>2</sup> � <sup>1</sup>

Graphical interpretations of the characteristics are shown in **Figure 8**.

Based on the above results, we can compile a table that compares the key properties of individual compensation methods (**Table 2**). In the evaluation, we consider a system operating to the optimal load at a distance corresponding to the coupling factor *k* < 0.1. The supply voltage is the same for all compensation

� �

0 0 *<sup>j</sup>*

*dt* � *<sup>R</sup>*2*i*<sup>3</sup> <sup>þ</sup>

*j* 1 *ωC*<sup>1</sup>

*<sup>R</sup>*<sup>1</sup> <sup>þ</sup> *<sup>j</sup> <sup>ω</sup>L*<sup>1</sup> � <sup>1</sup>

Unlike previous models, in this case it is necessary to compile four equations with four unknowns, the integrodifferential form is represented by the Eqs. (11).

> 1 *C*2 ð*t* 0

*di*<sup>2</sup> *dt* <sup>þ</sup> *<sup>M</sup> di*<sup>3</sup>

ð Þ *i*<sup>4</sup> � *i*<sup>3</sup> *dt* þ *uC*2ð Þ <sup>0</sup> ¼ 0

0 0

�*jωM* 0

*ωC*<sup>2</sup> � � *j*

*ωC*<sup>2</sup>

*ωC*<sup>2</sup>

� � � � �

(12)

*RZ* � *<sup>j</sup> ωC*<sup>2</sup>

*dt* <sup>¼</sup> <sup>0</sup>

(11)

The equations below for the calculation by the loop method can be expressed as (10).

$$
\begin{bmatrix}
\dot{I}\_{\rm S1} \\
\dot{I}\_{\rm S2} \\
\dot{I}\_{\rm S3}
\end{bmatrix} = \begin{bmatrix}
\frac{-j}{\dot{\rho}C\_1} & \frac{j}{\dot{\rho}C\_1} & 0 \\
\frac{j}{\dot{\rho}C\_1} & R\_1 + j\left(\dot{\rho}L\_1 - \frac{1}{\dot{\rho}C\_1}\right) & -j\dot{\rho}M \\
0 & -j\dot{\rho}M & R\_2 + R\_Z + j\left(\dot{\rho}L\_2 - \frac{1}{\dot{\rho}C\_2}\right)
\end{bmatrix} \begin{bmatrix}
\dot{U}\_1 \\
0 \\
0
\end{bmatrix} \tag{10}
$$

From the above equations we again obtain the courses of all-important operating variables, shown in **Figure 5**. As in the previous case, the system achieves maximum efficiency at the optimized load (*Rz* = 22 Ω), while the transmitted power is significantly lower compared to the achievable value (**Figure 6**).

#### **Figure 6.**

*Dependency of output power (left) and system efficiency (right) on frequency and power transfer distance for P-S compensation.*

*Theoretical and Practical Design Approach of Wireless Power Systems DOI: http://dx.doi.org/10.5772/intechopen.95749*

**Figure 7.**

�*u*<sup>1</sup> þ

*right – Circuit suited for loop current analysis.*

� 1 *C*1 ð*t* 0

�*L*<sup>2</sup> *di*<sup>3</sup> *dt* � *<sup>M</sup> di*<sup>2</sup>

�*j ωC*<sup>1</sup>

*j ωC*<sup>1</sup>

´*IS*1 ´*IS*2 ´*IS*3

**Figure 6.**

**48**

*P-S compensation.*

**Figure 5.**

2 6 4

1 *C*1 ð*t* 0

> *j ωC*<sup>1</sup>

icantly lower compared to the achievable value (**Figure 6**).

� �

*<sup>R</sup>*<sup>1</sup> <sup>þ</sup> *<sup>j</sup> <sup>ω</sup>L*<sup>1</sup> � <sup>1</sup>

ð Þ *i*<sup>1</sup> � *i*<sup>2</sup> *dt* þ *uC*1ð Þ <sup>0</sup> ¼ 0

*Simplified equivalent circuit for parallel - series compensation, left – Circuit with initial variables,*

*Wireless Power Transfer – Recent Development, Applications and New Perspectives*

*di*<sup>2</sup> *dt* <sup>þ</sup> *<sup>M</sup> di*<sup>3</sup>

0

�*jωM*

*dt* <sup>¼</sup> <sup>0</sup>

2 6 4

� � � � �

*ωC*<sup>2</sup>

� �

(9)

ð Þ *i*<sup>1</sup> � *i*<sup>2</sup> *dt* � *uC*1ð Þ <sup>0</sup> þ *R*1*i*<sup>2</sup> þ *L*<sup>1</sup>

*ωC*<sup>1</sup>

<sup>0</sup> �*jωM R*<sup>2</sup> <sup>þ</sup> *RZ* <sup>þ</sup> *<sup>j</sup> <sup>ω</sup>L*<sup>2</sup> � <sup>1</sup>

*dt* � *<sup>R</sup>*2*i*<sup>3</sup> � *RZi*<sup>3</sup> <sup>¼</sup> <sup>0</sup>

The equations below for the calculation by the loop method can be expressed as (10).

From the above equations we again obtain the courses of all-important operating variables, shown in **Figure 5**. As in the previous case, the system achieves maximum efficiency at the optimized load (*Rz* = 22 Ω), while the transmitted power is signif-

*Dependency of output power (left) and system efficiency (right) on frequency and power transfer distance for*

*Simplified equivalent circuit for parallel - parallel compensation, left – Circuit with initial variables, right – Circuit suited for loop current analysis.*

### **2.4 Parallel-parallel compensation**

Parallel–parallel compensation uses two resonant capacitors connected in parallel to both coupling coils. The circuit model can be seen in **Figure 7**.

Unlike previous models, in this case it is necessary to compile four equations with four unknowns, the integrodifferential form is represented by the Eqs. (11).

$$\begin{aligned} & -u\_1 + \frac{1}{C\_1} \int\_0^t (i\_1 - i\_2) dt + u\_{C\_1(0)} = 0 \\ & -\frac{1}{C\_1} \int\_0^t (i\_2 - i\_1) dt - u\_{C\_1(0)} + R\_1 i\_2 + L\_1 \frac{di\_2}{dt} + M \frac{di\_3}{dt} = 0 \\ & -L\_2 \frac{di\_3}{dt} - M \frac{di\_2}{dt} - R\_2 i\_3 + \frac{1}{C\_2} \int\_0^t (i\_4 - i\_3) dt + u\_{C\_2(0)} = 0 \\ & -\frac{1}{C\_2} \int\_0^t (i\_4 - i\_3) dt - u\_{C\_2(0)} - R\_2 i\_4 = 0 \end{aligned} \tag{11}$$

After stabilization we can rewrite the equations into the form (12).

$$
\begin{bmatrix}
\dot{I}\_{31} \\
\dot{I}\_{32} \\
\dot{I}\_{33} \\
\dot{I}\_{34} \\
\dot{I}\_{34}
\end{bmatrix} = \begin{bmatrix}
\frac{j}{\dot{o}\alpha\mathbf{C}\_{1}} & R\_{1} + j\left(\alpha\mathbf{L}\_{1} - \frac{1}{\alpha\mathbf{C}\_{1}}\right) & -j\alpha\mathbf{M} & 0 \\
0 & -j\alpha\mathbf{M} & R\_{2} + j\left(\alpha\mathbf{L}\_{2} - \frac{1}{\alpha\mathbf{C}\_{2}}\right) & \frac{j}{\alpha\mathbf{C}\_{2}} \\
0 & 0 & \frac{j}{\alpha\mathbf{C}\_{2}} & R\_{2} - \frac{j}{\alpha\mathbf{C}\_{2}}
\end{bmatrix} \begin{bmatrix}
\dot{U}\_{1} \\
0 \\
0 \\
0
\end{bmatrix} \tag{12}
$$

Graphical interpretations of the characteristics are shown in **Figure 8**.

#### **2.5 Overall comparison**

Based on the above results, we can compile a table that compares the key properties of individual compensation methods (**Table 2**). In the evaluation, we consider a system operating to the optimal load at a distance corresponding to the coupling factor *k* < 0.1. The supply voltage is the same for all compensation topologies and is not regulated.

already physically exists. On the other side the calculations offer the possibility of

Within next text, the systematic analytical procedure for calculation of key parameters of coupling elements for wireless power transfer is described, whereby the application area for any compensation technique can be considered here.

In practice, circular or spiral coils are most often used for high-frequency purposes. The reason is the high gradients of the electric field, which arise on all structural edges of the coil in the case of parallel resonance. These gradients significantly worsen the quality factor and thus the operating characteristics of the resulting system [20, 21]. The mutual inductance of different clusters of air coils of spiral shape can be based on the application of the analytical rule for the magnetic vector potential in

*cos <sup>φ</sup>*<sup>0</sup> ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

The resulting relationship is based on the direct application of Biot-Savart's law.

2 *kI* � *kI* � �*Κ*ð Þ� *kI*

Where *K(kI)* a *E(kI)* are elliptical integrals of first and second type and have

ð*π* 2 0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4*rr*<sup>1</sup>

The derived relations correspond to a simplified geometry, where only a coaxial arrangement is considered [22]. In order to be able to calculate the mutual inductance of the coils of general geometry and arrangement, we must introduce the

For this special case, the procedure for modifying the previous equations was indicated in [22, 23]. For the mutual inductance of the two turns from **Figure 9**

> *kII* ffiffi

*<sup>r</sup>*2*<sup>y</sup> cos*ð Þ *φ* h i <sup>1</sup> � *<sup>k</sup>*<sup>2</sup>

<sup>2</sup> <sup>þ</sup> *<sup>z</sup>* � *<sup>z</sup>*<sup>0</sup> ð Þ<sup>2</sup>

*II* 2

� �*K k*ð Þ� *II E k*ð Þ*II* h i*d<sup>φ</sup>*

<sup>3</sup> <sup>p</sup> , (17)

ð Þ *r* þ *r*<sup>1</sup>

In technical practice, these integrals are abundant, and therefore considerable attention has been paid to their enumeration in the past. The literature defines three basic types of these (elliptic) integrals, which can be combined with each other and

> 2 ffiffiffiffiffiffi *r*1*r* p

<sup>1</sup> � <sup>2</sup>*rr*<sup>1</sup> *cos <sup>φ</sup>*<sup>0</sup> ð Þþ *<sup>z</sup>* � <sup>z</sup><sup>0</sup> ð Þ<sup>2</sup> <sup>q</sup> *<sup>d</sup>φ*<sup>0</sup>

> 2 *kI Ε*ð Þ *kI* � � (14)

> > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>k</sup>*<sup>2</sup>

� � q

*<sup>I</sup> sin* <sup>2</sup> ð Þ *φ* , (13)

*dφ:* (15)

(16)

optimization within design procedure.

*DOI: http://dx.doi.org/10.5772/intechopen.95749*

*Theoretical and Practical Design Approach of Wireless Power Systems*

**3.1 Mutual inductance of planar coils**

cylindrical coordinates (13).

*Aφ*½ �¼ *r*, *φ*, *z*

easily converted to any special case.

ð*π* 2 0

possibility of deflection, see **Figure 9** left.

ffiffiffiffiffiffiffiffiffi *r*1*ir*2*<sup>j</sup>* p

Ð *π*

following forms:

*K k*ð Þ¼ *<sup>I</sup>*

*Mij* <sup>¼</sup> <sup>2</sup>*μ*<sup>0</sup> *π*

**51**

*Aφ*½ �¼ *r*, *φ*, *z*

*μ*0*I* <sup>4</sup>*<sup>π</sup> <sup>r</sup>*<sup>1</sup> ð*π* 0

The Eq. (14) was determined based on Eq. (13).

*dφ* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � *<sup>k</sup>*<sup>2</sup>

*<sup>I</sup> sin* <sup>2</sup> ð Þ *φ* <sup>q</sup> *aE k*ð Þ¼ *<sup>I</sup>*

Module of these integrals is determined using (16).

*kI* ¼

(left) we can write according to **Figure 9** (right) next Eq. (17).

<sup>0</sup> *cos*ðÞ � *<sup>d</sup>*

s

*μ*0*I* <sup>4</sup>*<sup>π</sup> <sup>r</sup>*<sup>1</sup>

*r*<sup>2</sup> þ *r*<sup>2</sup>

#### **Figure 8.**

*Dependency of output power (left) and system efficiency (right) on frequency and power transfer distance for P–P compensation.*


#### **Table 2.**

*Comparisons of key attributes of individual compensation techniques.*

Serial-to-series compensation acts as a current source over the monitored operating distance range, allowing higher power to be delivered to the load compared to other topologies. The disadvantage is the minimum overlap of work areas with maximum system performance and efficiency.

Serial-parallel compensation in this case does not bring any significant operational benefits, the only difference lies in the higher values of the optimal load. Unlike the previous solution, the circuit acts as a voltage source.

Parallel–series compensation offers partial overlap of work areas with maximum performance and efficiency. However, the theoretically achievable transmitted power values are significantly lower compared to the two previous configurations. The circuit has a voltage output and is much less sensitive to frequency detuning.

Parallel–parallel compensation offers current output with better coverage of areas of maximum power and efficiency along with better frequency stability. However, the transmitted powers are very low, as with parallel–series compensation.
