**4. Electric and magnetic fields**

**Figure 5** shows the situation when an electric and magnetic field component is emitted from the infinitesimal small element *dz* located at the lightning channel in the height *z*. The field emitted from the lightning channel arrives after the time interval *r/c* at the point X in the distance *s*. Thus, the retarded time *tx* is introduced to consider the propagation time:

$$t\_{\infty} = t - \frac{r}{c} = t - \frac{\sqrt{s^2 + z^2}}{c} \tag{34}$$

For an observer in point *X*, the apparent height of the lightning channel is *hx* = *vtx* (*v* = const.). Using *z* = *hx* in Eq. (34), the apparent height follows to [14]:

$$h\_x(t) = \frac{v}{c^2 - v^2} \left( c^2 t - \sqrt{v^2 c^2 t^2 + c^2 s^2 - v^2 s^2} \right) \tag{35}$$

The differentiation by time provides the apparent return stroke velocity [14]:

$$v\_x(t) = \frac{dh\_x(t)}{dt} = \frac{v}{1 - \frac{v^2}{c^2}} \left(1 - \frac{v^2 t}{\sqrt{v^2 c^2 t^2 + c^2 s^2 - v^2 s^2}}\right) \tag{36}$$

**Figure 5.** *Field emission from the return stroke channel.*

The electric field is perpendicular to the earth's surface. With the apparent height *hx* and the retarded time *tx*, the vertical electric field in the distance *s* is given by [14, 15]:

$$E(t) = E\_Q(t) + E\_i(t) + E\_{di}(t) + \Delta E(t) \tag{37}$$

The first term (*EQ*) represents the electrostatic field, the second term (*Ei*) the intermediate field, and the third term (*Edi*) the radiation field. The terms are given by (*ε<sup>0</sup>* = 8.854 *F/m*):

$$E\_Q(t) = \frac{1}{2\pi\epsilon\_0} \int\_0^{t\_x} \frac{s^2 - 2z^2}{r^5} \left( \int\_{t\_{x/0}}^{t\_x} i(z, \tau) d\tau \right) dz \tag{38}$$

$$E\_i(t) = \frac{1}{2\pi\varepsilon\_0} \int\_0^{\hbar\_x} \frac{z^2 - 2z^2}{cr^4} i(z, t\_x) dz \tag{39}$$

$$E\_{di}(t) = \frac{1}{2\pi\varepsilon\_0} \int\_0^{\hbar\_x} \frac{s^2}{c^2 r^3} \frac{\partial i(z, t\_x)}{\partial t} dz \tag{40}$$

In Eq. (38), the lower constant of integration *tx/0* denotes the retarded time when the tip of the lightning channel arrived at the height *z*. The magnetic field is horizontal to the earth's surface. The horizontal magnetic field is given by [14, 15]:

$$H(\mathbf{t}) = H\_i(\mathbf{t}) + H\_{di}(\mathbf{t}) + \Delta H(\mathbf{t})\tag{41}$$

The first term (Hi) represents the induction field, and the second term (*Hdi*) the radiation field. The terms are given by:

$$H\_i(t) = \frac{1}{2\pi} \int\_0^{\hbar\_x} i(z, t\_x) dz\tag{42}$$

$$H\_{di}(t) = \frac{1}{2\pi} \int\_0^{\hbar\_x} \frac{\partial i(z, t\_x)}{\partial t} dz \tag{43}$$

From **Figure 4**, it can be seen that the current changes abruptly at the upper end of the lightning channel. The abrupt current change represents a discontinuity at the open end of a transmission line. The discontinuity moves upwards with the return stroke velocity *v* = *dh/dt*, i.e., the current *i*(*h, t*) is turned on at the channel segment *dh* during the time interval *dt*. This is taken into account by the following term:

$$\frac{\partial \dot{i}(h,t)}{\partial t} dh = \frac{\partial h}{\partial t} di = -v \cdot di \tag{44}$$

The negative value indicates that the propagation of the current is in the opposite direction compared to the coordinate z. The abrupt current change is responsible for the additional far field terms *ΔE*(*t*) and *ΔH*(*t*) in Eq. (37) and in Eq. (41). These terms are often referred to in the literature as turn-on terms [16].

Of course, for an observer in the distance *s,* the real height has to be substituted by the apparent height *hx* and the real return stroke velocity by the apparent return stroke velocity *vx*.

*Return Stroke Process Simulation Using TCS Model DOI: http://dx.doi.org/10.5772/intechopen.98898*

The turn-on terms are finally given by [14]:

$$\begin{split} \Delta E(t) &= \frac{1}{2\pi\varepsilon\_{0}} \int\_{h\_{x,-}^{+}}^{h\_{x,+}^{2}} \frac{\delta i(z, t\_{x})}{c^{2}r^{3}} dh\_{x} \\ &= \ > \Delta E(t) = \frac{1}{2\pi\varepsilon\_{0}} \frac{s^{2}}{c^{2}\left(\sqrt{s^{2} + h\_{x}^{2}}\right)^{3}} v\_{x} \cdot i(h\_{x}, t\_{x}) \\ \Delta H(t) &= \frac{1}{2\pi} \int\_{h\_{x,-}^{-}}^{h\_{x,+}} \frac{s}{cr^{2}} \frac{\partial i(z, t\_{x})}{\partial t} dh\_{x} \\ &= \ > \Delta H(t) = \frac{1}{2\pi} \frac{s}{c(s^{2} + h\_{x}^{2})} v\_{x} \cdot i(h\_{x}, t\_{x}) \end{split} \tag{46}$$

### **5. Examples**

The waveform of the electric and magnetic fields is well-known from measurements at various distances. According to the distance, the field is usually classified into three groups: near field, intermediate field, and far field. The near field distance range is up to several kilometers, the intermediate field distance range is from several kilometers up to several tens of kilometers, and the far field distance range is from several tens of kilometers up to several hundreds of kilometers.

The basic features are as follows [17]:


The following shows, by using two examples, that the TCS model reproduces these basic features.

The first example analyses the influence of the ground reflection on the current and on the electric and magnetic field for a typical negative first return stroke at a near distance. The second example presents the electric and magnetic field for a typical subsequent return stroke at a near, intermediate, and far distance. In both examples, the return stroke velocity is chosen to *v* = *c*/3 = 100 *m/μs*. The following predefined source current is used (For example, see [12]):

$$i\_Q(t) = \frac{i\_{Q/max}}{\eta} \cdot \frac{\left(\frac{t}{\tau\_1}\right)^m}{\mathbf{1} + \left(\frac{t}{\tau\_1}\right)^m} \cdot e^{t/\tau\_2} \tag{47}$$

The coefficient *η* denotes the correction factor for the current maximum. The coefficients *τ<sup>1</sup>* and *τ<sup>1</sup>* are the front and decay time parameters of the current waveform.

#### **5.1 Influence of ground reflection**

In Eq. (47), the current parameters are chosen to *iQ/max* = 30 *kA*, *m* = 5, *τ<sup>1</sup>* = 1.26 *μs* and *τ<sup>2</sup>* = 56.3 *μs*. The maximum current steepness (*diQ*/*dtmax*) is about 32 *kA/μs*. These values are typical for a negative first return stroke [18, 19].

The characteristic impedance of the lightning channel is about 1000 Ω [20]. The grounding resistance of poorly grounded buildings is often in the same order of magnitude. In this case, the ground reflection can be neglected. On the other hand, the current ground reflections cannot be ignored when well-grounded structures have much lower resistances. For instance, the well-grounded Peissenberg tower has a ground reflection coefficient of about *ρ* = 0.7 [21].

In the following, the two cases are analyzed, i.e., the ground reflection coefficient is set to *ρ* = 0 and *ρ* = 0.7. **Figure 6a** and **b** show the influence of the ground reflection on the channel-base currents (*iBase*) (at the striking point). For *ρ* = 0, the peak current is identical with the peak value of the source current, *iQ/max* = 30 *kA*. For *ρ* = 0.7, the peak current is 43.5 *kA*, equating to an increase of 45%. **Figure 6c** shows, that the maximum current steepness also got higher by 65.6%, from 24.1 *kA/ μs* for *ρ* = 0 to 39.9 *kA/μs* for *ρ* = 0.7. The 10–90% rise time decreased accordingly, from *T*10–90% ≈ 1.4 *μs* for *ρ* = 0 to *T*10–90% ≈ 1.1 *μs* for *ρ* = 0.7.

**Figures 7** and **8** show the corresponding electric and magnetic fields in a distance of 3 *km*. The measured Initial Peak of the electric field (*Emax*) and magnetic field (*Hmax*) is successfully reproduced, but it is more pronounced for *ρ* = 0.7, i.e., the initial field peak is about 44% higher for *ρ* = 0.7 compared to *ρ* = 0.

The electric field exhibits the Ramp (**Figure 7a**), and the magnetic field exhibits the Hump (**Figure 8a**), known from measurements. The Ramp and Hump are more pronounced for *ρ* = 0.7 compared to *ρ* = 0. The different steepness of the Ramp is due to the current reflections, but the final value of the electric field is the same (not shown here) because the total charge transfer is unaltered.

**Figure 6.**

*Channel-base current as a function of the ground reflection factor ρ, showing (a) the total current, (b) the current front, and (c) the current derivative.*

**Figure 7.** *Electric field in 3* km *distance, showing (a) total field and (b) field front.*

*Return Stroke Process Simulation Using TCS Model DOI: http://dx.doi.org/10.5772/intechopen.98898*

**Figure 8.** *Magnetic field in 3 km distance, showing (a) total field and (b) field front.*

**Figure 9** *Derivative of (a) Electric field and (b) magnetic field in 3* km *distance.*

**Figure 9** shows that the influence of the current reflections on the field derivative is comparably low. For *ρ* = 0.7, the maximum of the electric and magnetic field derivative (*dE/dtmax*, *dH/dtmax*) is about 26% higher, and the full width at half maximum (*FWHM*) is higher by less than 20%.

#### **5.2 Electric and magnetic field at near, intermediate, and far distance**

In Eq. (47), the current parameters are chosen to *iQ/max* = 10 *kA*, *m* = 4, *τ<sup>1</sup>* = 0.7 *μs* and *τ<sup>2</sup>* = 30 *μs*. Ground reflection is ignored (*ρ* = 0) and the return stroke velocity is chosen to 100 *m/μs*. For the channel-base current (*iBase*) (at the striking point), the peak value is 10 *kA*, the 10–90% rise time is 0.94 *μs* and the time to half value is 31 *μs*. These values are typical for subsequent return strokes [18, 19, 22].

**Figure 10** shows the electric and magnetic fields at 1 *km* (near field), 10 *km* (intermediate field), and 100 *km* (far field). It can be seen that the main characteristics of the electric and magnetic fields are reproduced with the TCS model, i.e., the Initial Peak of the electric and magnetic field, the Ramp of the near electric field, the Hump of the near magnetic field, and the Zero Crossing of the electric and magnetic far field.

At far distances, the electric and magnetic field is approximately given by the radiation term (*Edi*, *Hdi*) according to Eqs. (40) and (43). In this case, the electromagnetic field (*Efar*, *Hfar*) has a behavior like a plane wave in free space, given by the following formula (Compare [6]):

$$\frac{E\_{far}}{H\_{far}} = \mathcal{c} \cdot \mu\_0 = \Gamma\_0 \tag{48}$$

*<sup>μ</sup><sup>0</sup>* = 4<sup>π</sup> �10�<sup>7</sup> *H/m*: Permeability of free space. *Γ<sup>0</sup>* = π �120 Ω ≈ 377 Ω: Impedance of free space.

**Figure 10.**

*Electric field,* E*(*t*), and magnetic field,* H*(*t*), of a subsequent return stroke in the distances of 1, 10, and 100* km*. For the channel-base current (at the striking point), the peak current, 10* kA*, rise time, 0.94* μs*, and time to half value, 31* μs*.*

As shown in Eq. (48), the electric and the magnetic far fields are linked together by the impedance of free space. Therefore, the waveform of the electric and magnetic fields is the same at far distances, as shown in **Figure 10**.

At 100 *km*, the Initial Peak of the electric field is 3.2 *V/m*, which agrees very well with measured data that varies between 2.7 and 5.0 *V/m* [5].

#### **5.3 Summery**

The examples show that the main features of the measured electric and magnetic fields are reproduced with the TCS model. These are the Ramp of the near electric field, Hump of the near magnetic field, Zero Crossing of the far distant electric and magnetic fields, and Initial Peak of the electric and magnetic fields for near, intermediate, and far distances.

*Return Stroke Process Simulation Using TCS Model DOI: http://dx.doi.org/10.5772/intechopen.98898*
