*3.5.1 Series parasitic resistance of spiral planar coil*

In this approximation, we will only talk about coils wound with a copper conductor of circular cross-section. These with their shape most closely resemble parts of the Archimedean spiral, where regarding **Figure 9** on the right we can denote the inner radius as *r1A* and the outer radius as *r1B*. The distance between the individual turns *e* and the number of turns N will be constant. As mentioned earlier, the key length here is played primarily by the length of the conductor of the wound coil. We can write an equation for the Archimedean spiral in polar coordinates

$$r = r\_{1A} + \zeta \cdot \rho,\\
2\pi\zeta = e.\tag{38}$$

The length of the spiral thus described can be determined by integration (39)

$$l\_c = \int\_0^{2\pi N} \sqrt{\left(r\_{1A} + \frac{e}{2\pi}\rho\right)^2 + \left(\frac{dr}{d\rho}\right)^2} d\rho,\tag{39}$$

However, the disadvantage remains the fact that the integral (39) cannot be solved analytically. It is therefore necessary to integrate numerically for the calculation. It is also possible to use an approximation relation for an approximate calculation

$$l\_c = \frac{1}{2} \left[ \frac{e}{2\pi} (2\pi N)^2 + 2\pi N \left( \sqrt{r\_{1A}^2 + \left(\frac{e}{2\pi}\right)^2} + r\_{1A} \right) \right],\tag{40}$$

or simplification by means of an average radius, see (41). The calculation is then very fast and convenient.

$$l\_c = 2\pi N \frac{r\_{1A} + r\_{1B}}{2} \tag{41}$$

In addition, high-frequency applications require winding of a solid conductor to reduce the parasitic capacitance of the coil. Therefore, if we consider the effect of the skin effect, we can adjust (37) to the shape (44) for a conductor with radius *rv*. Here for *lc* we use one of the equations (39)–(41).

$$R\_{c-AC}(w) = \begin{cases} \rho\_{Cu} \frac{l\_c}{\pi \delta (2r\_v - \delta)}, \land \delta \le r\_v\\ \rho\_{Cu} \frac{l\_c}{\pi r\_v^2} \left(1 + 0.0021 \left(\frac{r\_v}{\delta}\right)^4\right), \land \delta > r\_v \end{cases}, \delta = \sqrt{\frac{2\rho\_{Cu}}{a\mu\_0}}\tag{42}$$

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

#### *3.5.2 Series parasitic resistance of rectangular planar coil*

**3.5 Series parasitic resistance of the coil**

general relation for resistance according to (37).

*3.5.1 Series parasitic resistance of spiral planar coil*

*lc* ¼

*e*

Here for *lc* we use one of the equations (39)–(41).

*ρCu lc πr*<sup>2</sup> *v*

8 >>><

>>>:

*lc* <sup>¼</sup> <sup>1</sup> 2

very fast and convenient.

*Rc*�*AC*ð Þ¼ *ω*

**58**

ð<sup>2</sup>*π<sup>N</sup>* 0

s

<sup>2</sup>*<sup>π</sup>* ð Þ <sup>2</sup>*π<sup>N</sup>* <sup>2</sup> <sup>þ</sup> <sup>2</sup>*π<sup>N</sup>*

The series resistance of the coil is one of the most critical parameters of the system with the greatest influence on its operational efficiency and it is therefore very important to know this value as accurately as possible. We can start with the

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

*R* ¼ *ρ l*

So far, we will not consider temperature or frequency dependences. While the effective area of the conductor S depends only on the current load, the length l already depends on the geometric shape of the coil. As mentioned earlier, spiral planar coils of solid conductor are more suitable for high frequency applications.

In this approximation, we will only talk about coils wound with a copper conductor of circular cross-section. These with their shape most closely resemble parts of the Archimedean spiral, where regarding **Figure 9** on the right we can denote the inner radius as *r1A* and the outer radius as *r1B*. The distance between the individual turns *e* and the number of turns N will be constant. As mentioned earlier, the key length here is played primarily by the length of the conductor of the wound coil. We

The length of the spiral thus described can be determined by integration (39)

*e* <sup>2</sup>*<sup>π</sup> <sup>φ</sup>* � �<sup>2</sup>

However, the disadvantage remains the fact that the integral (39) cannot be solved analytically. It is therefore necessary to integrate numerically for the calculation. It is

r

or simplification by means of an average radius, see (41). The calculation is then

In addition, high-frequency applications require winding of a solid conductor to reduce the parasitic capacitance of the coil. Therefore, if we consider the effect of the skin effect, we can adjust (37) to the shape (44) for a conductor with radius *rv*.

*lc*

*δ*

*πδ*ð Þ <sup>2</sup>*rv* � *<sup>δ</sup>* , <sup>∧</sup>*δ*≤*rv*

, ∧*δ*>*rv*

, *δ* ¼

*lc* <sup>¼</sup> <sup>2</sup>*π<sup>N</sup> <sup>r</sup>*1*<sup>A</sup>* <sup>þ</sup> *<sup>r</sup>*1*<sup>B</sup>*

*ρCu*

<sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*<sup>0021</sup> *rv*

� �<sup>4</sup> � �

" # !

*r*2 <sup>1</sup>*<sup>A</sup>* þ

*r*1*<sup>A</sup>* þ

also possible to use an approximation relation for an approximate calculation

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

*e* 2*π* � �<sup>2</sup>

þ *r*1*<sup>A</sup>*

<sup>2</sup> (41)

ffiffiffiffiffiffiffiffiffi 2*ρCu ωμ*<sup>0</sup>

(42)

s

can write an equation for the Archimedean spiral in polar coordinates

*<sup>S</sup>* (37)

*r* ¼ *r*1*<sup>A</sup>* þ *ζ* � *φ*, 2*πζ* ¼ *e:* (38)

*dφ*, (39)

, (40)

*dr dφ* � �<sup>2</sup>

The coil has rectangular turns to achieve maximum inductance (**Figure 16**). If we denote the external dimensions of the coil by the letters a and b and consider the spacing between the individual turns *e* constant, we can write Eq. (43) for the resulting resistance.

$$R\_{co-DC}(\theta) = \frac{\rho(\theta)}{\mathcal{S}\_{\text{Cu}}} \left( 2 \sum\_{i=1}^{N} [a + b - [1 + (i - 1)4]e] \right) \tag{43}$$

Furthermore, if we choose an insulated RF cable with wires whose diameter is much smaller than the penetration depth *d*, we can certainly rule out the effect of the skin effect. We are then talking about *a* conductor with an effective cross section SCu, whose frequency dependence is caused only by the phenomenon of proximity. With a few modifications, it can be further simplified (44) by removing the summation into the shape

$$R\_{co-DC}(\theta) = \frac{\rho(\theta)}{\mathcal{S}\_{Cu}} 2N[a + b + e(1 - 2N)] \tag{44}$$

In all the cases described, the turns are evenly distributed in one layer with a constant *e*. The current flowing through the coil thus has the same direction in all turns and generates a magnetic field with lower intensity on the external turns and higher intensity on the internal turns. This magnetic field induces eddy currents into all coil turns and thus increases its overall resistance. This process is commonly referred to as the proximity phenomenon and can be conveniently calculated from the relationship for eddy current losses in individual parallel conductors. For one fiber of diameter *ds* of an insulated cable of length *l*, exposed to an external magnetic field *B* of a harmonic waveform of angular frequency *w*, we can write

$$R\_{\text{prox}} = \frac{\pi l d\_s^4}{64\rho} \omega^2 \mathcal{B}^2 \tag{45}$$

If we use a cable made of *nf* insulated conductors for winding of the coil, it is possible to adjust *eg* (44) to the shape (46) by counting (42).

$$R\_{cn-AC}(\theta,\omega) = \left(\frac{\rho(\theta)}{n\_t\frac{\pi d\_t^2}{4}} + n\_t\frac{\pi d\_t^4}{64\rho(\theta)}\alpha^2 B\_{\text{stiff}}^2\right) nN\left[a + \tan\left(\frac{\pi}{n}\right)(1-N)\varepsilon\right] \tag{46}$$

**Figure 16.** *Rectangular coil identification for the calculation of parasitic resistance.*

To illustrate, we will analyze the following geometry. We consider a square coil with an outer edge of length *a* = 500 mm and *N* = 20 turns. We choose the spacings between the turn's axes *e* = 8 mm. The maximum operating frequency is *f* = 300 kHz, while for a nominal current *I* = 5 A we choose a current value *J* = 7 MA/m<sup>2</sup> . The calculation is valid for a temperature of 20° C. **Figure 17** shows the dependence of the DC resistance from (44), the AC resistance from (46) and the total resistance. The minimum value of the number of RF wires is determined from (47) and corresponds to the rated current load (ns-min = 61).

$$n\_{s-\min} \doteq \frac{4I}{\pi d\_s^2 I} \tag{47}$$

*3.5.3 Series parasitic resistance of rectangular planar coil*

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

*Theoretical and Practical Design Approach of Wireless Power Systems*

quality factor such as

*I <sup>ω</sup>L*�*jR ωCp*ð Þ� *R*þ*jωL j* n o

*R <sup>ω</sup>L*�*jR ωCp*ð Þ� *R*þ*jωL j* n o

coil's own resonant frequency

*Q* ¼

**Figure 18.**

**61**

� � � � � �

The most general definition of the quality factor is based on the ratio of accumulated and lost energy in the investigated passive component. For AC supply we can write (50), where the influence of the electric field prevails in the case of a

capacitor and the influence of the magnetic field in the case of a coil.

*<sup>Q</sup>* <sup>¼</sup> *<sup>ω</sup> Wmg* � � � � P *j*

*C*2

also use the relation to calculate the quality factor

� � � � � � �

*Parasitic components of the coil and the quality factor characteristic.*

� � � � � � ¼ *<sup>Q</sup>* <sup>¼</sup> *<sup>ω</sup> Wmg* � *Wel* �

*<sup>ω</sup> <sup>L</sup>*�*CpR*<sup>2</sup>�*ω*<sup>2</sup>*CpL*<sup>2</sup> ð Þ <sup>1</sup>þ*ω*<sup>2</sup>*Cp CpR*<sup>2</sup> ½ � <sup>þ</sup>*<sup>L</sup>*ð Þ *<sup>ω</sup>*<sup>2</sup>*CpL*�<sup>2</sup> *R*

*<sup>p</sup>ω*2*R*<sup>2</sup>þð Þ *<sup>ω</sup>*<sup>2</sup>*CpL*�<sup>1</sup> <sup>2</sup>

The first part of the result of Eq. (52) corresponds to the quality factor of the individual coil, the second part then respects the effect of parasitic capacitance between the turns. A closer look reveals that there is a frequency at which both parts are equal and (52) gives zero result. This frequency is often referred to as the

*fr*�*self* <sup>¼</sup> <sup>1</sup>

quality factor and the character of the resulting reactance on the frequency is plotted for selected values of the parasitic capacity of the inductance and the series

2*π* ffiffiffiffiffiffiffiffi *LCp*

The situation is indicated in **Figure 18** on the right, where the dependency of the

� �

*P j*

If we consider an ideal coil (R-L circuit) without parasitic capacitance, we get a

¼ *ω* 1 <sup>2</sup> *LI*<sup>2</sup> *m* 1 <sup>2</sup> *RI*<sup>2</sup> *m*

For more complicated circuits, such as components with parasitic effects, we can

� � � � � � �

�

¼ 1 *R*

ffiffiffi *L C* r

<sup>¼</sup> *<sup>ω</sup><sup>L</sup>*

<sup>¼</sup> *<sup>ω</sup><sup>L</sup>*

� � � � *:* (50)

*<sup>R</sup>* , (51)

� � � � *:* (52)

*<sup>R</sup>* � *<sup>ω</sup>Cp <sup>R</sup>*<sup>2</sup> <sup>þ</sup> *<sup>ω</sup>*<sup>2</sup>*L*<sup>2</sup> � � *R*

<sup>p</sup> *:* (53)

By further increase of the number of wires, we therefore only increase the current possibilities of the coil.

As the number of wires increases, the DC resistance *RDC* decreases sharply, but the AC resistance *Rprox* also increases. The optimum can be found by solving Eq. (48).

$$\frac{d}{dn\_s} \left[ \left( \frac{\rho(\theta)}{n\_s \frac{\pi d\_s^2}{4}} + n\_s \frac{\pi d\_s^4}{64 \rho(\theta)} \alpha^2 B\_{\text{stiff}}^2 \right) nN \left[ a + \tan \left( \frac{\pi}{n} \right) (1 - N) e \right] \right] = 0 \tag{48}$$

The condition is met just when it applies

$$m\_s \doteq \frac{16\rho(\theta)\sqrt{a - e(1+N)\tan\left(\frac{\pi}{n}\right)}}{\pi\sqrt{a^2 B\_{\text{sir}}^2 d\_s^6 \left[a - e(1+N)\tan\left(\frac{\pi}{n}\right)\right]}} \cdot \tag{49}$$

After substituting, we get the value *ns* = 328 wires, which corresponds to the value in **Figure 17**.

**Figure 17.** *Nomogram for the calculation of the optimal number of litz wire.*

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

#### *3.5.3 Series parasitic resistance of rectangular planar coil*

To illustrate, we will analyze the following geometry. We consider a square coil with an outer edge of length *a* = 500 mm and *N* = 20 turns. We choose the spacings

. The calculation is valid for a temperature of 20° C. **Figure 17** shows the dependence of the DC resistance from (44), the AC resistance from (46) and the total resistance. The minimum value of the number of RF wires is determined from

> 4*I πd*<sup>2</sup>

*<sup>s</sup> <sup>J</sup>* (47)

3

5 ¼ 0 (48)

between the turn's axes *e* = 8 mm. The maximum operating frequency is *f* = 300 kHz, while for a nominal current *I* = 5 A we choose a current value *J* = 7

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

*ns*� *min* ≐

By further increase of the number of wires, we therefore only increase the

the AC resistance *Rprox* also increases. The optimum can be found by solving

1

h i <sup>2</sup>

*B*2 *st*r

16*ρ ϑ*ð Þ

*ω*<sup>2</sup>*B*<sup>2</sup> *st*r*d*<sup>6</sup>

As the number of wires increases, the DC resistance *RDC* decreases sharply, but

<sup>A</sup>*nN a* <sup>þ</sup> *tan <sup>π</sup>*

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi *<sup>a</sup>* � *<sup>e</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>N</sup> tan <sup>π</sup>*

*<sup>s</sup> <sup>a</sup>* � *<sup>e</sup>*ð Þ <sup>1</sup> <sup>þ</sup> *<sup>N</sup> tan <sup>π</sup>*

q � �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

After substituting, we get the value *ns* = 328 wires, which corresponds to the

*n* � �

ð Þ 1 � *N e*

*n*

*n* <sup>q</sup> � � � � *:* (49)

(47) and corresponds to the rated current load (ns-min = 61).

MA/m<sup>2</sup>

Eq. (48).

*d dns*

current possibilities of the coil.

*ρ ϑ*ð Þ *ns πd*<sup>2</sup> *s* 4

0 @

4

value in **Figure 17**.

**Figure 17.**

**60**

þ *ns*

The condition is met just when it applies

*ns* ≐

*Nomogram for the calculation of the optimal number of litz wire.*

*π*

*πd*<sup>4</sup> *s* <sup>64</sup>*ρ ϑ*ð Þ*ω*<sup>2</sup>

The most general definition of the quality factor is based on the ratio of accumulated and lost energy in the investigated passive component. For AC supply we can write (50), where the influence of the electric field prevails in the case of a capacitor and the influence of the magnetic field in the case of a coil.

$$Q = \alpha \frac{\left| \mathcal{W}\_{\rm mg} - \mathcal{W}\_{el} \right|}{P\_j} = \frac{1}{R} \sqrt{\frac{L}{C}}. \tag{50}$$

If we consider an ideal coil (R-L circuit) without parasitic capacitance, we get a quality factor such as

$$Q = \alpha \frac{\left| \mathcal{W}\_{mg} \right|}{\mathcal{P}\_j} = \alpha \frac{\frac{1}{2} L I\_m^2}{\frac{1}{2} R I\_m^2} = \frac{\alpha L}{R},\tag{51}$$

For more complicated circuits, such as components with parasitic effects, we can also use the relation to calculate the quality factor

$$Q = \left| \frac{I\left\{\frac{\alpha L - jR}{\alpha C\_p (R + j\alpha L) - j}\right\}}{R\left\{\frac{\alpha L - jR}{\alpha C\_p (R + j\alpha L) - j}\right\}} \right| = \left| \frac{\frac{\alpha \left(L - C\_p R^2 - \alpha^2 C\_p L^2\right)}{1 + \alpha^2 C\_p \left[C\_p R^2 + L\left(\alpha^2 C\_p L - 2\right)\right]}}{\frac{R}{C\_p \alpha^2 R^2 + \left(\alpha^2 C\_p L - 1\right)^2}} \right| = \left| \frac{\alpha L}{R} - \frac{\alpha C\_p \left(R^2 + \alpha^2 L^2\right)}{R} \right|. \tag{52}$$

The first part of the result of Eq. (52) corresponds to the quality factor of the individual coil, the second part then respects the effect of parasitic capacitance between the turns. A closer look reveals that there is a frequency at which both parts are equal and (52) gives zero result. This frequency is often referred to as the coil's own resonant frequency

$$f\_{r-\text{self}} = \frac{1}{2\pi\sqrt{LC\_p}}.\tag{53}$$

The situation is indicated in **Figure 18** on the right, where the dependency of the quality factor and the character of the resulting reactance on the frequency is plotted for selected values of the parasitic capacity of the inductance and the series

**Figure 18.** *Parasitic components of the coil and the quality factor characteristic.*

resistance of the coil (*Cp* = 5 pF, *L* = 0.1 mH, *R* = 1 Ω). As can be seen from the figure, when reaching the natural resonant frequency of the *fr-self circuit* »225 MHz, the quality factor is equal to zero and at the same time the inductive character of the reactance changes to capacitive character. For this reason, we always try to operate the coil at a frequency much lower than the self-resonant frequency.
