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.
- Return Stroke
- Electric Field
- Magnetic Field
- TCS model
- Ground Reflection
- Channel Top Reflection
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 . 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) .
The lightning current may contain several components, from which the so-called 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 ). 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, 7, 8, 9, 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:
In the equivalent circuit, the current injection can be represented by a current source
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 (
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
Also, in this case, the current injection can be represented by a current source
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 (
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 (
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 (
The current (
The current reflections are considered by the ground reflection coefficient
3.1 Top reflection coefficient R
Figure 4 shows the upward-moving current wave
The increasing lightning channel creates the new channel segment
In Eq. (4), the negative sign is due to the current propagation in the opposite direction compared to the coordinate
The current is given by:
For example, if we assume the return stroke velocity
3.2 Source current
iQand channel base current iBase
The TCS model uses the source current (
The channel-base current (
The reflected current component ρ·iBase/d(t) arrives at the top of the lightning channel after the delay time T. Because at the time (
After the reflection at the upper end of the lightning channel, the (reflected) current wave
At ground level, the downward moving current wave is given by:
Thus, Eq. (11) can be rewritten as:
Substituting the time
Now, the following iterative procedure is applied to Eq. (16). Between each iteration step, the equation is multiplied by the factor (
The addition of the series of equations gives the following formula:
Substituting the time
Now, Eq. (21) is multiplied by the factor (−
and from it
3.3 Lightning current along return stroke channel
When the downward propagating current wave,
In the opposite case, when an upward-moving current starts at the time (t-z/c) from ground level (
At the upper end of the return stroke channel (
With the channel height
3.4 Special case of no ground reflections
The reflections at the ground are often ignored, and the ground reflection coefficient is set to
From Eq. (25), the relation between the source current (
4. Electric and magnetic fields
Figure 5 shows the situation when an electric and magnetic field component is emitted from the infinitesimal small element
The differentiation by time provides the apparent return stroke velocity :
The first term (
In Eq. (38), the lower constant of integration
The first term (Hi) represents the induction field, and the second term (
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
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
Of course, for an observer in the distance
The turn-on terms are finally given by :
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 :
The electric and magnetic field exhibits an Initial Peak at distances of more than several hundred meters.
The near electric field exhibits a Ramp (up to the maximum) after the Initial Peak,
The near magnetic field exhibits a Hump after the Initial Peak,
The electric and magnetic far field exhibits a Zero Crossing after the Initial Peak.
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
5.1 Influence of ground reflection
In Eq. (47), the current parameters are chosen to
The characteristic impedance of the lightning channel is about 1000 Ω . 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
In the following, the two cases are analyzed, i.e., the ground reflection coefficient is set to
Figures 7 and 8 show the corresponding electric and magnetic fields in a distance of 3
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
Figure 9 shows that the influence of the current reflections on the field derivative is comparably low. For
5.2 Electric and magnetic field at near, intermediate, and far distance
In Eq. (47), the current parameters are chosen to
Figure 10 shows the electric and magnetic fields at 1
At far distances, the electric and magnetic field is approximately given by the radiation term (
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.
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.