**3.5 Series parasitic resistance of the coil**

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 general relation for resistance according to (37).

$$R = \rho \frac{l}{S} \tag{37}$$

*3.5.2 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*

*Rco*�*DC*ð Þ¼ *<sup>ϑ</sup> ρ ϑ*ð Þ

*SCu*

*Rco*�*DC*ð Þ¼ *<sup>ϑ</sup> ρ ϑ*ð Þ

possible to adjust *eg* (44) to the shape (46) by counting (42).

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

þ *ns*

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

2 X *N*

*SCu*

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

*Rprox* <sup>¼</sup> *<sup>π</sup>ld*<sup>4</sup>

*s* <sup>64</sup>*<sup>ρ</sup> <sup>ω</sup>*<sup>2</sup>

If we use a cable made of *nf* insulated conductors for winding of the coil, it is

*B*2 *st*r

1

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

*i*¼1

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

resulting resistance.

the summation into the shape

*Rcn*�*AC*ð Þ¼ *<sup>ϑ</sup>*,*<sup>ω</sup> ρ ϑ*ð Þ

**Figure 16.**

**59**

*ns πd*<sup>2</sup> *s* 4

0 @

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

> ½ � *a* þ *b* � ½ � 1 þ ð Þ *i* � 1 4 *e* !

> > 2*N a*½ � þ *b* þ *e*ð Þ 1 � 2*N* (44)

*B*<sup>2</sup> (45)

*n* � �ð Þ <sup>1</sup> � *<sup>N</sup> <sup>e</sup>* h i (46)

(43)

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.
