**3.1 Mutual inductance of planar coils**

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 cylindrical coordinates (13).

$$A\_{\varphi}[r,\varphi,z] = \frac{\mu\_0 I}{4\pi} r\_1 \Big|\_{0}^{\pi} \frac{\cos\left(\wp'\right)}{\sqrt{r^2 + r\_1^2 - 2rr\_1\cos\left(\wp'\right) + \left(z - z'\right)^2}} d\wp',\tag{13}$$

The resulting relationship is based on the direct application of Biot-Savart's law. 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 easily converted to any special case.

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

$$A\_{\varphi}[r,\varphi,z] = \frac{\mu\_0 I}{4\pi} r\_1 \frac{2}{\sqrt{r\_1 r}} \left[ \left(\frac{2}{k\_I} - k\_I\right) K(k\_I) - \frac{2}{k\_I} E(k\_I) \right] \tag{14}$$

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

$$K(k\_I) = \int\_0^\frac{d\rho}{\sqrt{1 - k\_I^2 \sin^2(\rho)}} aE(k\_I) = \int\_0^\frac{\pi}{2} \left[ \sqrt{1 - k\_I^2 \sin^2(\rho)} \right] d\rho. \tag{15}$$

Module of these integrals is determined using (16).

$$k\_I = \sqrt{\frac{4rr\_1}{\left(r+r\_1\right)^2 + \left(z-z'\right)^2}}\tag{16}$$

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

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

$$M\_{\rm ij} = \frac{2\mu\_0}{\pi} \sqrt{r\_{\rm II} r\_{\rm j}} \frac{\int\_0^\pi \left[\cos\left(\right) - \frac{d}{r\_{\rm I}} \cos\left(\rho\right)\right] \left[\left(1 - \frac{k\_{\rm II}^2}{2}\right) K(k\_{\rm II}) - E(k\_{\rm II})\right] d\rho}{k\_{\rm II} \sqrt{3}},\tag{17}$$

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

Source output type current voltage voltage Current Power transfer ability higher higher lower lower Max power and efficiency overlap No No Partial Partial Optimum load value lower higher lower higher Frequency sensitivity higher higher lower lower

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

*S-S S-P P-S P–P*

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.

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,

**3. Identification of key system parameters and analytical approach for**

In the previous chapter, the principal characteristics regarding basic modifications of the main circuits for wireless power systems were derived and described. All models, in some form, use concentrated parameters of spare electrical circuits. These parameters can be determined basically in two ways, i.e. by calculation and by measurement, while the measurement can be used only if the analyzed system

the transmitted powers are very low, as with parallel–series compensation.

maximum system performance and efficiency.

*Comparisons of key attributes of individual compensation techniques.*

**Table 2.**

**50**

**Figure 8.**

*P–P compensation.*

**design of coupling elements**

Unlike the previous solution, the circuit acts as a voltage source.

*Criteria Compensating topology*

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

#### **Figure 9.**

*Displacement of coil's turns and presentation of the situation for determination of mutual inductance for circular coils.*

where

$$\sqrt{1-\sin^2(\)\cos^2(\boldsymbol{\rho})+\left(\frac{d}{r\_{2j}}\right)^2-\frac{2d}{r\_{2j}}\cos\left(\right)\cos\left(\boldsymbol{\rho}\right)},\tag{18}$$

Thanks to the equivalent replacement of individual turns with concentric rectangles/squares, we are able to solve the magnetic field around the coil relatively easily and analytically. **Figure 11** indicates the relative position of two coils of

Let us now focus on the i-th turn of the lower coil and the j-th turn of the upper coil. The magnetic field passing through the upper coil (excited by the lower coil) can be calculated from (21), where Biz is the induction of the magnetic field in the z-axis.

*BizdS <sup>j</sup>* ¼

ð *S j*

*Bcos*ð Þ*θ dS <sup>j</sup>* (21)

different dimensions and number of turns spaced by a length z.

*Real (left) and simplified (right) geometry of the coil with rectangular shape.*

*Theoretical and Practical Design Approach of Wireless Power Systems*

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

*Simplified situation for determination of mutual inductance between rectangular coils.*

*ϕij* ¼ ð *S j B* ! *idS* ! *<sup>j</sup>* ¼ ð *S j*

**Figure 11.**

**53**

**Figure 10.**

and then

$$k\_{\rm II} = \sqrt{\frac{4\frac{r\_{\rm i}}{r\_{\rm i}}}{\left(1 + \frac{r\_{\rm i}}{r\_{\rm i}}\right)^2 + \left(\frac{x}{r\_{\rm i}} - \frac{r\_{\rm i}}{r\_{\rm i}}\cos\left(\right)\sin\left(\rho\right)\right)^2}}\tag{19}$$

Both coils have one or more turns, and since the equations derived above apply only to the arrangement of simple loops, it is not possible to apply them directly. The calculation is divided into *N1 N2* sub-steps, where the mutual inductance of all turn's combinations of the first and second coil is determined. Substituting (17) into (20) we get the total mutual inductance.

$$\mathcal{M} = \sum\_{i=1}^{N\_1} \sum\_{j=1}^{N\_2} \mathcal{M}\_{ij} \tag{20}$$

As mentioned above, spiral-shaped coils are rather used for high-frequency applications, while rectangular or square-shaped coils are suitable for applications operating at lower frequencies. On the one hand, there are no electric field gradients, the coupling is rather inductive, and on the other hand we try to make maximum use of the built-up areas to maximize the coupling factor between the coils. As in the previous case, Biot-Savart's law can be applied here as well.

However, since it is not a circular coil, the advantages of the cylindrical coordinate system cannot be used, and the calculation is considerably complicated. To avoid confusing relationships, we will only consider the coaxial arrangement of two coils. These can have different geometries and different numbers of turns.

**Figure 10** shows the real and simplified geometry of the coil, on which the derivation of the calculations will be performed. As can be seen in the figure on the left (**Figure 10**), the actual turns have different lengths at the same position, making the whole arrangement asymmetrical. The analytical solution of the field would then be very complicated and quite confusing.

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

**Figure 10.** *Real (left) and simplified (right) geometry of the coil with rectangular shape.*

Thanks to the equivalent replacement of individual turns with concentric rectangles/squares, we are able to solve the magnetic field around the coil relatively easily and analytically. **Figure 11** indicates the relative position of two coils of different dimensions and number of turns spaced by a length z.

Let us now focus on the i-th turn of the lower coil and the j-th turn of the upper coil. The magnetic field passing through the upper coil (excited by the lower coil) can be calculated from (21), where Biz is the induction of the magnetic field in the z-axis.

$$\boldsymbol{\phi}\_{\vec{\boldsymbol{\eta}}} = \int\_{\mathcal{S}\_{\boldsymbol{j}}} \overrightarrow{\boldsymbol{B}}\_{i} d\overrightarrow{\boldsymbol{S}}\_{\boldsymbol{j}} = \int\_{\mathcal{S}\_{\boldsymbol{j}}} \boldsymbol{B}\_{\text{i}\boldsymbol{x}} d\mathbf{S}\_{\boldsymbol{j}} = \int\_{\mathcal{S}\_{\boldsymbol{j}}} \boldsymbol{B} \cos(\boldsymbol{\theta}) d\mathbf{S}\_{\boldsymbol{j}} \tag{21}$$

**Figure 11.** *Simplified situation for determination of mutual inductance between rectangular coils.*

where

**Figure 9.**

*circular coils.*

and then

**52**

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*Displacement of coil's turns and presentation of the situation for determination of mutual inductance for*

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

<sup>þ</sup> *<sup>z</sup> r*1*i* � *<sup>r</sup>*2*<sup>j</sup>*

*<sup>M</sup>* <sup>¼</sup> <sup>X</sup> *N*<sup>1</sup>

coils. These can have different geometries and different numbers of turns. **Figure 10** shows the real and simplified geometry of the coil, on which the derivation of the calculations will be performed. As can be seen in the figure on the left (**Figure 10**), the actual turns have different lengths at the same position, making the whole arrangement asymmetrical. The analytical solution of the field

would then be very complicated and quite confusing.

*i*¼1

As mentioned above, spiral-shaped coils are rather used for high-frequency applications, while rectangular or square-shaped coils are suitable for applications operating at lower frequencies. On the one hand, there are no electric field gradients, the coupling is rather inductive, and on the other hand we try to make maximum use of the built-up areas to maximize the coupling factor between the coils. As in the previous case, Biot-Savart's law can be applied here as well.

However, since it is not a circular coil, the advantages of the cylindrical coordinate system cannot be used, and the calculation is considerably complicated. To avoid confusing relationships, we will only consider the coaxial arrangement of two

X *N*<sup>2</sup>

*j*¼1

Both coils have one or more turns, and since the equations derived above apply only to the arrangement of simple loops, it is not possible to apply them directly. The calculation is divided into *N1 N2* sub-steps, where the mutual inductance of all turn's combinations of the first and second coil is determined. Substituting (17) into

*d r*2*j* � �<sup>2</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 *<sup>r</sup>*2*<sup>j</sup> r*1*i*

� <sup>2</sup>*<sup>d</sup> r*2*j*

*<sup>r</sup>*1*<sup>i</sup> cos*ðÞ*sin* ð Þ *φ* � �<sup>2</sup> vuuut *:* (19)

*cos*ðÞ*cos*ð Þ *φ*

*Mij* (20)

, (18)

ðÞ*cos* <sup>2</sup>ð Þþ *φ*

<sup>1</sup> <sup>þ</sup> *<sup>r</sup>*2*<sup>j</sup> r*1*i* � �<sup>2</sup>

<sup>1</sup> � *sin* <sup>2</sup>

*kII* ¼

(20) we get the total mutual inductance.

s

Furthermore, we can use Biot-Savart's law to determine the increment of the BCD magnetic field from the current-carrying segment of the i-th turn as:

$$\overrightarrow{B}\_{\rm CD} = \frac{\mu\_0}{4\pi} \frac{1}{\sqrt{\left(b\_i - y\right)^2 + x^2}} \left[ \frac{a\_i + x}{\sqrt{\left(b\_i - y\right)^2 + x^2 + \left(a\_i + x\right)^2}} + \frac{a\_i - x}{\sqrt{\left(b\_i - y\right)^2 + x^2 + \left(a\_i - x\right)^2}} \right], \tag{22}$$

An if applicable next equation:

$$\cos\left(\theta\right) = \frac{b\_i - y}{\sqrt{\left(b\_i - y\right)^2 + z^2}},\tag{23}$$

enough to swap ai with bi and cj with dj in (26). Due to the symmetry, the Eq. (28)

And because in the case of a unity current considering mutual inductance

*<sup>M</sup>* <sup>¼</sup> <sup>X</sup> *N*<sup>1</sup>

*DAVGK*<sup>1</sup> *ln <sup>K</sup>*<sup>2</sup>

*<sup>p</sup>* <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> � *<sup>D</sup>*<sup>1</sup> *D*<sup>2</sup> þ *D*<sup>1</sup> *i*¼1

X *N*<sup>2</sup>

*j*¼1

To calculate the intrinsic inductance of a planar coil, it is possible to find simple approximation relations, which are suitable for consequent mathematical derivations. However, their big disadvantage is only an approximate calculation with an often-indeterminate error. In addition, the relationships apply only to coils with an

> *p* � �

, *DAVG* <sup>¼</sup> *<sup>D</sup>*<sup>2</sup> <sup>þ</sup> *<sup>D</sup>*<sup>1</sup>

Here, for p, the turn's filling factor on the coil surface and DAVG is represented as

In **Table 3**, the coefficients depending on the approximated coil geometry are calculated (**Figure 12**). The coefficients *K1 - K4* must always be selected according

*Coil shape K***<sup>1</sup>** *K***<sup>2</sup>** *K***<sup>3</sup>** *K***<sup>4</sup>** *Circular* 1 2.46 0 0.2 *Squared* 1.27 2.07 0.18 0.13 *Hexagonal* 1.09 2.23 0 0.17 *Octagonal* 1.07 2.29 0 0.19

<sup>þ</sup> *<sup>K</sup>*3*<sup>p</sup>* <sup>þ</sup> *<sup>K</sup>*4*p*<sup>2</sup> � � (31)

between the i-th and j-th turns next equation is valid (29)

*Theoretical and Practical Design Approach of Wireless Power Systems*

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

**3.2 Self-inductance of planar coils**

*La* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup> 2 *N*2

*Estimated coefficients for identification of the shape of planar coil.*

*Allowed degenerations of the coil's geometry for calculation of inductance using Eq. (31).*

the total mutual inductance of both coils is based on (30).

*Ψ AB*�*<sup>z</sup>* ¼ *ΨCD*�*za Ψ DA*�*<sup>z</sup>* ¼ *ΨBC*�*<sup>z</sup>* (28)

*Mij* (30)

<sup>2</sup> (32)

*Mij Ψij*�*<sup>z</sup>* ¼ *Ψ AB*�*<sup>z</sup>* þ *ΨBC*�*<sup>z</sup>* þ *ΨCD*�*<sup>z</sup>* þ *Ψ DA*�*z*, (29)

will apply.

equilateral plan

the mean winding diameter.

to the current geometry.

**Table 3.**

**Figure 12.**

**55**

For z-component of BCD induction we can derive (24)

$$B\_{\rm CD-x} = \frac{\mu\_0}{4\pi} \frac{b\_i - y}{\left(b\_i - y\right)^2 + x^2} \left[ \frac{a\_i + x}{\sqrt{\left(b\_i - y\right)^2 + x^2 + \left(a\_i + x\right)^2}} + \frac{a\_i - x}{\sqrt{\left(b\_i - y\right)^2 + x^2 + \left(a\_i - x\right)^2}} \right]. \tag{24}$$

For the total coupled magnetic flux with CD segment, we can write integral in form of (25)

$$
\Psi\_{\text{CD}-x} = \int\_{-d\_j}^{d\_j} dy \int\_{-c\_j}^{c\_j} B\_{\text{CD}-x} d\mathfrak{x}.\tag{25}
$$

As shown in [24], although the solution of the integral (25) is more complicated, we obtain a purely analytical relation (26).

$$\begin{aligned} \Psi\_{CD-x} &= \frac{\mu\_0}{4\pi} \left[ \Gamma\_1 - \left( a\_i + c\_j \right) \tanh^{-1} \left( \frac{a\_i + c\_j}{\Gamma\_1} \right) - \Gamma\_2 \right] \\ &+ \left( a\_i - c\_j \right) \tanh^{-1} \left( \frac{a\_i - c\_j}{\Gamma\_2} \right) - \Gamma\_3 \\ &+ \left( a\_i + c\_j \right) \tanh^{-1} \left( \frac{a\_i + c\_j}{\Gamma\_3} \right) - \Gamma\_4 \\ &- \left( a\_i - c\_j \right) \tanh^{-1} \left( \frac{a\_i - c\_j}{\Gamma\_4} \right) - \Gamma\_4 \end{aligned} \tag{26}$$

In relation (26) it is still necessary to substitute substitution (27).

$$\begin{aligned} \Gamma\_1 &= \sqrt{\left(b\_i + d\_j\right)^2 + \mathbf{z}^2 + \left(a\_i + c\_j\right)^2} \\ \Gamma\_2 &= \sqrt{\left(b\_i + d\_j\right)^2 + \mathbf{z}^2 + \left(a\_i - c\_j\right)^2} \\ \Gamma\_3 &= \sqrt{\left(b\_i - d\_j\right)^2 + \mathbf{z}^2 + \left(a\_i + c\_j\right)^2} \\ \Gamma\_4 &= \sqrt{\left(b\_i - d\_j\right)^2 + \mathbf{z}^2 + \left(a\_i - c\_j\right)^2} \end{aligned} \tag{27}$$

The magnetic flux from the other segments (AB, BC and DA) can be easily determined using the same relations. For example, to calculate the segment BC, it is *Theoretical and Practical Design Approach of Wireless Power Systems DOI: http://dx.doi.org/10.5772/intechopen.95749*

enough to swap ai with bi and cj with dj in (26). Due to the symmetry, the Eq. (28) will apply.

$$
\Psi\_{AB-x} = \Psi\_{CD-x} a \,\Psi\_{DA-x} = \Psi\_{BC-x} \tag{28}
$$

And because in the case of a unity current considering mutual inductance between the i-th and j-th turns next equation is valid (29)

$$M\_{\vec{\text{ij}}} \Psi\_{\vec{\text{ij}}-x} = \Psi\_{AB-x} + \Psi\_{BC-x} + \Psi\_{CD-x} + \Psi\_{DA-x},\tag{29}$$

the total mutual inductance of both coils is based on (30).

$$\mathcal{M} = \sum\_{i=1}^{N\_1} \sum\_{j=1}^{N\_2} \mathcal{M}\_{ij} \tag{30}$$

### **3.2 Self-inductance of planar coils**

Furthermore, we can use Biot-Savart's law to determine the increment of the

*ai* <sup>þ</sup> *<sup>x</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*cos*ð Þ¼ *<sup>θ</sup> bi* � *<sup>y</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *bi* � *y*

*ai* <sup>þ</sup> *<sup>x</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

q þ

<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *ai* <sup>þ</sup> *<sup>x</sup>*

For the total coupled magnetic flux with CD segment, we can write integral in

As shown in [24], although the solution of the integral (25) is more complicated,

� � *tanh* �<sup>1</sup> *ai* <sup>þ</sup> *cj*

*Γ*2 � �

*Γ*3 � �

*Γ*4 � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

� �<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>þ</sup> *ai* <sup>þ</sup> *cj*

� �<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>þ</sup> *ai* � *cj*

� �<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>þ</sup> *ai* <sup>þ</sup> *cj*

� �<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>þ</sup> *ai* � *cj*

The magnetic flux from the other segments (AB, BC and DA) can be easily determined using the same relations. For example, to calculate the segment BC, it is

ð*<sup>d</sup> <sup>j</sup>* �*d <sup>j</sup> dy* ð*<sup>c</sup> <sup>j</sup>* �*c <sup>j</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *ai* <sup>þ</sup> *<sup>x</sup>* <sup>2</sup> q þ *ai* � *<sup>x</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*ai* � *<sup>x</sup>* ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *ai* � *<sup>x</sup>*

<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup> <sup>þ</sup> ð Þ *ai* � *<sup>x</sup>* <sup>2</sup>

3 7 5,

(22)

2

(24)

(26)

(27)

3 7 5*:*

ð Þ *bi* � *y*

<sup>q</sup> , (23)

ð Þ *bi* � *y*

*BCD*�*zdx:* (25)

� *Γ*<sup>2</sup>

q

<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup>

2

q

*Γ*1 � �

� *Γ*<sup>3</sup>

� *Γ*<sup>4</sup>

� *Γ*<sup>4</sup> �

� �<sup>2</sup>

� �<sup>2</sup>

� �<sup>2</sup>

� �<sup>2</sup>

BCD magnetic field from the current-carrying segment of the i-th turn as:

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

ð Þ *bi* � *y*

For z-component of BCD induction we can derive (24)

ð Þ *bi* � *y*

*ΨCD*�*<sup>z</sup>* ¼

�

þ *ai* � *cj*

þ *ai* þ *cj*

� *ai* � *cj*

q

q

q

q

*Γ*<sup>1</sup> ¼

*Γ*<sup>2</sup> ¼

*Γ*<sup>3</sup> ¼

*Γ*<sup>4</sup> ¼

<sup>4</sup>*<sup>π</sup> <sup>Γ</sup>*<sup>1</sup> � *ai* <sup>þ</sup> *cj*

� � *tanh* �<sup>1</sup> *ai* � *cj*

� � *tanh* �<sup>1</sup> *ai* <sup>þ</sup> *cj*

� � *tanh* �<sup>1</sup> *ai* � *cj*

In relation (26) it is still necessary to substitute substitution (27).

*bi* þ *d <sup>j</sup>*

*bi* þ *d <sup>j</sup>*

*bi* � *d <sup>j</sup>*

*bi* � *d <sup>j</sup>*

*B* ! *CD* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup> 4*π*

*BCD*�*<sup>z</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

form of (25)

**54**

4*π*

1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ *bi* � *y*

An if applicable next equation:

*bi* � *y* ð Þ *bi* � *y*

<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup>

we obtain a purely analytical relation (26).

*<sup>Ψ</sup>CD*�*<sup>z</sup>* <sup>¼</sup> *<sup>μ</sup>*<sup>0</sup>

2 6 4

q

<sup>2</sup> <sup>þ</sup> *<sup>z</sup>*<sup>2</sup>

2 6 4

> To calculate the intrinsic inductance of a planar coil, it is possible to find simple approximation relations, which are suitable for consequent mathematical derivations. However, their big disadvantage is only an approximate calculation with an often-indeterminate error. In addition, the relationships apply only to coils with an equilateral plan

$$L\_{\rm at} = \frac{\mu\_0}{2} N^2 D\_{AVG} K\_1 \left[ \ln \left( \frac{K\_2}{p} \right) + K\_3 p + K\_4 p^2 \right] \tag{31}$$

Here, for p, the turn's filling factor on the coil surface and DAVG is represented as the mean winding diameter.

$$p = \frac{D\_2 - D\_1}{D\_2 + D\_1}, D\_{AVG} = \frac{D\_2 + D\_1}{2} \tag{32}$$

In **Table 3**, the coefficients depending on the approximated coil geometry are calculated (**Figure 12**). The coefficients *K1 - K4* must always be selected according to the current geometry.


**Table 3.**

*Estimated coefficients for identification of the shape of planar coil.*

**Figure 12.** *Allowed degenerations of the coil's geometry for calculation of inductance using Eq. (31).*

**Figure 13.** *Time-varying electromagnetic field around the system of coupling coils.*

### **3.3 Coupling coefficient**

The magnetic coupling between two coils is formed by a magnetic field, which is generated by a transmitting coil. For many reasons, this array can never be coupled to the receiving coil in its full size, and the larger the array, the better the coupling is achieved. This phenomenon is described by the so-called coupling factor *k*, which takes values from 0 to 1. The coupling factor depends on the geometry of the coils and their mutual position. Many authors mistakenly qualify the degree of coupling of two coils based on the shape of the electromagnetic field in their surroundings. This approach leads to misinterpretations mainly because the field itself is variable in time and looks different at different times. The situation is clearly shown in **Figure 13**.

For example, in S-S compensation, the currents flowing through the primary and secondary windings are time-shifted by 90 electrical degrees. While at the instant *j* = 0° and *j* = 50° the coils appear to be coupled, while at the instant *j* = 135° they are without mutual coupling according to the shape of the field (**Figure 13**). The only reliable way to determine the coupling factor is to apply relation (33).

$$k = \frac{M}{\sqrt{L\_1 L\_2}}\tag{33}$$

**3.4 Parasitic capacitance of the coil**

determined as

**Figure 14.**

*radial displacement (right).*

be seen in **Figure 15**.

**Figure 15.**

**57**

between individual turns e we can write the relation (34).

*Theoretical and Practical Design Approach of Wireless Power Systems*

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

*CCp* <sup>¼</sup> <sup>X</sup>

*D2* = 2r2, we get the modification (35) in the form (36).

*CCp* <sup>¼</sup> <sup>X</sup>

0 @ 0 @

*N*�1

*i*¼1

*Situation of the coil's turn placement for the calculation of parasitic capacitance.*

For the capacitance between two turns with mean radius *ri* and the distance

*Dependency of coupling factor on the geometrical properties of the coil and mutual distance (left) and mutual*

*πε*0*ri ln <sup>e</sup> rv*

� � (34)

*:* (35)

(36)

*Cip* <sup>¼</sup> <sup>3</sup> 2

If we have a coil with *N* turns, the total parasitic capacitance must be

3 2

Further to the pattern of **Figure 12** to the left we denote the outer diameter

0 @

4

*πε*0*ri ln <sup>e</sup> rv* � �

> *ln <sup>e</sup> rv* � �

*πε*0½ � *r*<sup>2</sup> � ð Þ *rv* þ *e=*2 *i*

The geometric arrangement, according to which (36) can be easily applied, can

Unlike high-frequency systems, at lower operating frequencies, ropes with insulated conductors are used almost exclusively. The reason is the lower influence of parasitic capacitances and especially the better current utilization of the coil.

2 �<sup>1</sup>

1 A 3 5

1 A

�1

3 5

1 A

�1

*N*�1

*i*¼1

2 3

2 4

We will explain some relationships on a simple example, in which we determine the coupling factor of two coaxially placed coils of circular shape with planar design. We will perform the calculation on three similar geometries (**Figure 12** left), where the first pair of coils will have an inner diameter *D1* = 100 mm and an outer diameter *D2* = 200 mm, whereby the other two pairs of coils will have the same dimensions multiplied by two and three. The number of turns is the same *N* = 5 for all cases. The coupling factor will be plotted for a distance *z* = 5 ÷ 300 mm.

As shown in **Figure 14** (left), changing the distance of the coils, the coupling factor *k* decreases rapidly, while the rate of this decrease is highly dependent on the respective geometry of the coils. It is therefore better to choose larger coil dimensions to improve the coupling at higher distances. In **Figure 14** (right) we see the effect of the misalignment of the coils in the *x*-axis at their constant distance in the *z*-axis. The coupling factor is somewhat less sensitive to this method of deflection.

**Figure 14.**

**3.3 Coupling coefficient**

*Time-varying electromagnetic field around the system of coupling coils.*

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

**Figure 13**.

**56**

**Figure 13.**

The magnetic coupling between two coils is formed by a magnetic field, which is generated by a transmitting coil. For many reasons, this array can never be coupled to the receiving coil in its full size, and the larger the array, the better the coupling is achieved. This phenomenon is described by the so-called coupling factor *k*, which takes values from 0 to 1. The coupling factor depends on the geometry of the coils and their mutual position. Many authors mistakenly qualify the degree of coupling of two coils based on the shape of the electromagnetic field in their surroundings. This approach leads to misinterpretations mainly because the field itself is variable in time and looks different at different times. The situation is clearly shown in

For example, in S-S compensation, the currents flowing through the primary and secondary windings are time-shifted by 90 electrical degrees. While at the instant *j* = 0° and *j* = 50° the coils appear to be coupled, while at the instant *j* = 135° they are without mutual coupling according to the shape of the field (**Figure 13**). The only reliable way to determine the coupling factor is to apply relation (33).

> *<sup>k</sup>* <sup>¼</sup> *<sup>M</sup>*ffiffiffiffiffiffiffiffiffiffi *L*1*L*<sup>2</sup>

the first pair of coils will have an inner diameter *D1* = 100 mm and an outer diameter *D2* = 200 mm, whereby the other two pairs of coils will have the same dimensions multiplied by two and three. The number of turns is the same *N* = 5 for

all cases. The coupling factor will be plotted for a distance *z* = 5 ÷ 300 mm.

As shown in **Figure 14** (left), changing the distance of the coils, the coupling factor *k* decreases rapidly, while the rate of this decrease is highly dependent on the respective geometry of the coils. It is therefore better to choose larger coil dimensions to improve the coupling at higher distances. In **Figure 14** (right) we see the effect of the misalignment of the coils in the *x*-axis at their constant distance in the *z*-axis. The coupling factor is somewhat less sensitive to this method of deflection.

We will explain some relationships on a simple example, in which we determine the coupling factor of two coaxially placed coils of circular shape with planar design. We will perform the calculation on three similar geometries (**Figure 12** left), where

p (33)

*Dependency of coupling factor on the geometrical properties of the coil and mutual distance (left) and mutual radial displacement (right).*

### **3.4 Parasitic capacitance of the coil**

For the capacitance between two turns with mean radius *ri* and the distance between individual turns e we can write the relation (34).

$$C\_{ip} = \frac{3}{2} \frac{\pi \varepsilon\_0 r\_i}{\ln\left(\frac{\varepsilon}{r\_v}\right)}\tag{34}$$

If we have a coil with *N* turns, the total parasitic capacitance must be determined as

$$\mathbf{C\_{Cp}} = \left(\sum\_{i=1}^{N-1} \left[ \left(\frac{3}{2} \frac{\pi \varepsilon\_0 r\_i}{\ln\left(\frac{\varepsilon}{r\_v}\right)}\right)^{-1} \right] \right)^{-1} . \tag{35}$$

Further to the pattern of **Figure 12** to the left we denote the outer diameter *D2* = 2r2, we get the modification (35) in the form (36).

$$\mathbf{C\_{Cp}} = \left(\sum\_{i=1}^{N-1} \left[\frac{2}{3} \frac{\ln\left(\frac{\varepsilon}{r\_v}\right)}{\pi \varepsilon\_0 [r\_2 - (r\_v + e/2)i]}\right] \right)^{-1} \tag{36}$$

The geometric arrangement, according to which (36) can be easily applied, can be seen in **Figure 15**.

Unlike high-frequency systems, at lower operating frequencies, ropes with insulated conductors are used almost exclusively. The reason is the lower influence of parasitic capacitances and especially the better current utilization of the coil.

**Figure 15.** *Situation of the coil's turn placement for the calculation of parasitic capacitance.*
