Return Stroke Process Simulation Using TCS Model

*Fridolin Heidler*

### **Abstract**

The Traveling Current Source (TCS) model describes the electrical processes during the lightning return stroke phase. The TCS model assumes that the lightning current is injected at the top of the increasing return stroke channel represented by a transmission line. The electric and magnetic field is calculated based on the spatial and temporal distribution of the lightning current along the return stroke channel. It is shown that the main characteristics of the measured electric and magnetic fields can be reproduced with the TCS model. These are the Initial Peak of the electric and magnetic fields for near intermediate and far distances, the Ramp (up to the maximum) of the near electric field, the Hump of the near magnetic field after the initial peak, and the Zero Crossing of the far distant electric and magnetic fields. The fundamentals of the model are presented, and the model is extended to consider the current reflections occurring at the ground and the upper end of the return stroke channel. To this end, the ground reflection factor ρ and the top reflection factor R are introduced. Due to the increasing return stroke channel, the top reflection factor is a function of the return stroke velocity. The total current is composed of the source current according to the TCS model and the reflected currents. It is shown that the ground reflection causes significant variation in the waveform of the channel-base current and the electric and magnetic fields.

**Keywords:** Return Stroke, Lightning, Electric Field, Magnetic Field, Simulation, TCS model, Ground Reflection, Channel Top Reflection

#### **1. Introduction**

The threat of lightning can be classified into two separate groups, given by the direct and the indirect effects. The direct effects include physical losses due to the hot lightning channel and the high lightning current. Typical direct effects are mechanical damage, fire ignition, and the life-threatening hazard by lightning impact to persons. The basic protection measures against this threat are installing air termination systems, down conductor systems, and grounding systems [1, 2].

In contrast, the indirect effects are caused by nearby lightning events. Typical indirect effects are over-voltages which affect the electric and electronic systems and devices. The over-voltages are caused by partial currents that enter the structure and the coupling effects due to the high electric and magnetic fields radiated by lightning.

Meanwhile, the economic losses caused by the indirect effects are much higher compared to the direct effects [3]. This is attributed to the widespread use of electrical and electronic systems and devices in private buildings and industrial

facilities. Countermeasures require the integration of lightning protection into the rules of electromagnetic compatibility (EMC) [4].

The lightning current may contain several components, from which the socalled return stroke current represents the highest threat. The return stroke current is a short current pulse, which lasts some tens to some hundreds of microseconds and may have an amplitude up to more than 100 kA (For example, see [5]). The currents generate electric and magnetic fields, which may be so intense that they couple over-voltages of several kilo-volts into installations inside buildings.

Examining these over-voltages requires simulation models that consider the return stroke process, including the electric and magnetic fields. To this end, return stroke models were developed which calculate the electric and magnetic field from the spatial and temporal distribution of the lightning current along the return stroke channel [6–10]. In these models, current reflections at the ground are commonly ignored. For this reason, the so-called traveling current (TCS)-model [11, 12] was developed, which considers the current reflections at the striking point.

One key task of the EMC is to evaluate the maximum threat which the electrical equipment and systems have to withstand. In the case of lightning, the electric and magnetic fields are highest if the orientation of the lightning channel is perpendicular to the earth's surface. For this reason, the lightning channel is considered with vertical orientation.

### **2. Physical background on TCS model**

Most of the observed cloud-to-earth flashes are of negative polarity. For this reason, the TCS model is presented for the negative return stroke. The return stroke phase involves two periods, the initial connecting leader period, followed by the second period when the downward leader channel is discharged.

#### **2.1 Connecting leader period**

The negative cloud-to-ground lightning starts with processes in which charges are separated and rearranged inside the thundercloud. Due to these processes, negative charges are accumulated, and the center of the negative charge is built up in the lower part of the thundercloud. When the accumulated charge exceeds a critical value, a negative leader is formed, propagating from the negative charge center towards the ground.

The hot core of the leader is surrounded by negative charges, which also move down. When the downward propagating leader comes close to the ground, the electric field increases due to the charge approach. Then, a connecting leader starts from the ground as soon as the electric field exceeds a critical value.

The electric field at the tip of the connecting leader is so high that charge carriers are separated by impact and photoionization around the leader tip. The electric field accelerates the charge carriers, and they move to the tip of the connecting leader. In this way, a current is injected at the tip of the connecting leader, shown in **Figure 1a**. The injected current is given by:

$$\vec{i}(h) = \int\_{A} \vec{j} \, d\vec{A}, \text{with } \vec{j} = \vec{Q}''' \cdot \vec{b} \cdot \vec{E} \tag{1}$$

*Q*000: Charge density of the charge carriers. *b*: Electrical mobility of the charge carriers. *E*: Electric field.

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

**Figure 1.**

*Assumption for the connecting leader period, showing (a) the physical model and (b) the electrical equivalent circuit.*

*j*: Current density.

*h*: Height of the upper end of the connecting leader.

In the equivalent circuit, the current injection can be represented by a current source *iQ* = *i* (*h*) located at the tip of the connecting leader in the height h. The current source travels at the connecting leader's upper end, which increases with the velocity *v*, shown in **Figure 1b**.

A certain time period is needed to separate the charge carriers and for the thermal ionization process at the tip of the connecting leader. For this reason, the traveling velocity (*v*) is less than the speed of light (*c*). On the other hand, the lower section of the connecting leader is already ionized, and this section represents a more or less good electrical conductor. Therefore, it is assumed that injected current propagates from the connecting leader tip with the speed of light to the ground.

#### **2.2 Discharge process of downward leader channel**

After contacting the upward propagating connecting leader with the downward leader, the negatively-charged shell of the downward leader is discharged, shown in **Figure 2a**. The charge carriers stored in the volume *dV* are injected into the tip of the lightning channel at the height *h* during the time interval *dt*. The current is given by:

$$i(h) = \frac{dQ}{dt} = \int\_{V} \frac{dQ'''}{dt}dV\tag{2}$$

Also, in this case, the current injection can be represented by a current source *iQ* = *i*(*h*) located at the tip of the increasing return stroke channel. **Figure 2b** shows the electrical equivalent circuit with the current source, which travels at the tip of the increasing return stroke channel towards the thundercloud.

**Figure 2.**

*Assumption for the discharge process of the downward leader channel, showing (a) physical model and (b) electrical equivalent circuit.*

A certain time period is needed to collect the charge carriers and the thermal ionization to form a new section of the return stroke. Therefore, in this case, the traveling velocity (*v*) is less than the speed of light (*c*).

#### **2.3 Summary**

The return stroke process consists of the initial connecting leader process and the subsequent discharge process of the downward leader channel. In the electrical equivalent circuit, both processes can be represented by a current source traveling from the ground in the direction of the thundercloud with the return stroke velocity (*v*). Therefore, in the TCS model, it is not necessary to distinguish between both processes.

#### **3. Current on the return stroke channel**

**Figure 3** shows the basic assumptions of the TCS model: The return stroke channel is perpendicular to the earth's surface, and it increases in the z-direction with constant return stroke velocity (*v*). The return stroke channel is considered as an (ideal) transmission line where the current pulses propagate with the speed of light (*c*). The ground is taken into account by an ideal-conducting plane.

The current (*iQ*) is injected at the top of the return stroke channel from the current source, which begins to travel at *t* = 0 from ground level. When injected current arrives at the ground, a fraction of the current is reflected depending on the ground resistance. The reflected current moves up, and it is reflected again at the end of the lightning channel.

The current reflections are considered by the ground reflection coefficient *ρ* and the top reflection coefficient R. The current at the altitude *z*, *i*(*z, t*), is composed by *Return Stroke Process Simulation Using TCS Model DOI: http://dx.doi.org/10.5772/intechopen.98898*

**Figure 3** *Current distribution on the return stroke channel.*

**Figure 4.** *Current reflection at the upper end of the return stroke channel.*

the downward moving current wave *id*(*z, t*) and the upward-moving current wave *iu*(*z, t*).
