Section 1
Two Practical Questions
This part answers two engineering questions about a line arrester during a lightning stroke. Question 1 — how much current flows through the arrester? When lightning hits a tower, ground wire or phase conductor, the surge current splits between the footing resistance, the ground wire, the phase conductor, the arrester and the adjacent spans. The arrester current sets the electrical stress on the arrester.
Question 2 — how much energy must the arrester absorb? The arrester carries current and absorbs energy:
\[ W \approx \int v_A(t)\,i_A(t)\,dt \;\approx\; E_A\,I_A\,T \]
Table 1 — Notation used in the arrester energy estimate.
| Symbol | Meaning |
|---|
| \(W\) | Arrester energy |
| \(v_A(t),\ i_A(t)\) | Instantaneous arrester voltage and current |
| \(E_A\) | Arrester discharge voltage |
| \(I_A\) | Approximate arrester current |
| \(T\) | Equivalent time constant / duration of the arrester current |
So the method first estimates the current, then its duration, then combines them into an energy.
This is Part Two of the line arrester series, continuing from the application and failure-probability foundations of Part One.
What this page teaches
- why two analyses (travelling-wave + inductance) are combined;
- how the lightning current divides — and why arrester current ≠ stroke current;
- why stroke location matters more than stroke magnitude;
- how arrester energy is estimated and why more towers reduce it;
- why a phase stroke with no ground wire is the most severe case;
- why arresters on every second tower may not protect the intermediate tower;
- how coupling, footing resistance and wood structures change protection;
- why strokes to the ground wire are far easier to protect than phase strokes.
Section 2
Two Analyses, Combined
Two complementary analyses are used — one for the fast front, one for the long tail:
Table 2 — Travelling-wave and inductance analyses and what each yields.
| Analysis | Used for | What It Gives |
|---|
| Travelling-wave | The fast front (first few microseconds) | Crest arrester current, crest tower / phase voltage, insulation voltage, adjacent-tower reflections |
| Inductance / time-constant | The longer current tail (energy) | The decay time of the arrester current — a practical energy estimate |
Lightning creates very steep waves that travel along the conductors at nearly the speed of light; the travelling-wave method captures the first microseconds. For the longer tail, the line is simplified to an inductance, so the current decay time can be estimated without modelling every reflection.
Section 3
The Key Parameters
Table 3 — Notation for the current, energy and travelling-wave parameters.
| Symbol | Meaning |
|---|
| \(I\) / \(i_A\) | Lightning stroke current / arrester current |
| \(i_g\) (or \(i_R\)) | Current through the ground / footing resistance |
| \(R_0\) / \(R_i\) | Low-current / impulse (high-current) footing resistance |
| \(Z_c\) / \(Z_g\) / \(Z_m\) | Phase, ground-wire and mutual surge impedances |
| \(E_A\) / \(E_0\) / \(R_A\) | Arrester discharge voltage / reference (intercept) voltage / dynamic resistance |
| \(T_s\) / \(T\) | Span travel time / lightning-current tail time constant |
| CFO / \(C\) | Critical flashover voltage / coupling factor (ground wire to phase) |
The most influential parameters are the lightning current magnitude and steepness, the tower footing resistance, the surge impedances, the arrester discharge voltage, the span length and the insulation CFO.
Current is a travelling-wave result
The arrester current is a travelling-wave result, not simply the lightning stroke current — it depends on where the stroke lands and how the surge divides and reflects.
Table 4 — How stroke location sets the current path and design concern.
| Event | Main Current Path | Main Concern |
|---|
| Stroke to phase conductor | Direct surge into the phase and the arrester path | Arrester current and energy at the struck phase |
| Stroke to ground wire / tower | Tower voltage and coupling to the phases | Backflashover prevention and arrester sharing |
| Intermediate-tower exposure | Reflections from protected adjacent towers | Possible flashover at the unprotected tower |
Section 4
Stroke to the Ground Wire: Current Division
Lightning hits the ground wire at an arrester tower. The ground wire is above the phase conductor, so coupling induces voltage on the phase, and the arrester operates to protect the phase insulation. Taking the ground wire and phase as infinitely long (ignoring adjacent-tower reflections for a first estimate), the entering current splits into the footing resistance, both directions along the ground wire, the arrester, and the coupled conductors. For \(I = 100\ \text{kA}\):
Table 5 — Current division values for a 100 kA ground-wire stroke.
| Quantity | Value |
|---|
| Lightning current \(I\) | \(100\ \text{kA}\) |
| Footing resistance \(R_0\) / impulse \(R_i\) | \(55\ \Omega\) / \(\approx 24.9\ \Omega\) |
| Arrester current \(i_A\) | \(\approx 6.7\ \text{kA}\) |
| Ground current \(i_g\) | \(\approx 82\ \text{kA}\) |
Arrester current is a small fraction
A \(100\ \text{kA}\) stroke to the ground wire gives only about \(6\)–\(7\ \text{kA}\) through the arrester — most current flows through the tower / ground path and along the ground wire.
Section 5
Why the Current Is Then Reduced
The median time to crest of a \(100\ \text{kA}\) stroke is about \(2.4\ \mu\text{s}\), while a \(230\ \text{m}\) span travels in about \(0.77\ \mu\text{s}\). The surge therefore reaches adjacent towers before the lightning current peaks, and reflections from adjacent spans reduce the arrester current. A correction factor is applied:
\[ K_{cp} \approx 0.90 \qquad\Rightarrow\qquad i_A:\ 6.7\ \text{kA} \;\to\; \approx 6\ \text{kA} \]
Hand equations are estimates, not exact
ATP simulation gives \(4.7\)–\(7.22\ \text{kA}\), so the hand calculation (\(\approx 6\ \text{kA}\)) is reasonable. Travelling-wave hand equations are useful for first engineering estimates but not exact — final design still needs EMTP®.
Section 6
Energy for a Stroke to the Ground Wire
Energy needs the current duration. The time constant is controlled by the footing resistance, the ground-wire surge behaviour and the span travel time, giving about \(10.4\ \mu\text{s}\) here. The resulting energy:
Table 6 — Arrester energy for a ground-wire stroke by tower count.
| Arrester Towers | Energy |
|---|
| 1 tower with arresters | \(13.9\ \text{kJ}\) |
| 11 towers with arresters | \(33\ \text{kJ}\) |
The simple method gives a value between these and is good enough for a first estimate — but detailed EMTP® simulation is still needed for the final design.
Section 7
Stroke to the Phase Conductor (with Ground Wire)
Now lightning hits the phase conductor directly at an arrester tower on a line with a ground wire or neutral — usually a shielding failure, whose current is lower because only smaller currents statistically bypass the shield wire. For a \(12.3\ \text{kA}\) shielding-failure current the calculated arrester current is about \(10.5\ \text{kA}\) (ATP: \(8.9\)–\(9.5\ \text{kA}\)). Compare the two stroke locations:
Table 7 — Arrester current compared for ground-wire and phase strokes.
| Case | Stroke Current | Arrester Current |
|---|
| Stroke to ground wire | \(100\ \text{kA}\) | \(\approx 6\ \text{kA}\) |
| Stroke to phase conductor | \(12.3\ \text{kA}\) | \(\approx 9\)–\(10\ \text{kA}\) |
Location matters more than magnitude
A direct phase stroke makes the arrester one of the main paths limiting the phase-to-ground voltage, so the arrester current is high even though the stroke current is small. Arrester duty depends not only on lightning current magnitude but on where the lightning hits.
Section 8
Energy for the Phase-Conductor Stroke
To estimate the longer tail, the line is replaced by inductances. The arrester voltage and current are related nonlinearly:
\[ E_A = E_{A2}\left(\frac{i_A}{I_2}\right)^{1/a} \]
For a metal-oxide arrester the discharge voltage rises only slowly as the current increases — the exponent \(a\) captures this strong nonlinearity. Between \(5\) and \(10\ \text{kA}\) the study uses \(a = 7.63\). The calculated energy is conservative compared with ATP:
Table 8 — Simulated arrester energy for a shielded phase stroke by tower count.
| Arrester Towers | ATP Energy |
|---|
| 3 towers with arresters | \(\approx 95\ \text{kJ}\) |
| 11 towers with arresters | \(\approx 51\ \text{kJ}\) |
More towers share the duty
More towers with arresters reduce the energy duty per arrester — the surge is shared among more arresters.
Section 9
Stroke to the Phase Conductor, No Ground Wire
This is the most severe case: lightning hits the phase conductor on a line with no ground wire or neutral. The arrester becomes the main protective path. For \(I = 100\ \text{kA}\) and a phase surge impedance \(Z_c = 366\ \Omega\), the arrester current is about \(85.7\ \text{kA}\) (ATP: \(\approx 80\ \text{kA}\)) — and the energy is enormous:
Table 9 — Simulated arrester energy for a phase stroke with no ground wire.
| Arrester Towers | ATP Energy |
|---|
| 3 towers with arresters | \(1733\ \text{kJ}\) |
| 11 towers with arresters | \(750\ \text{kJ}\) |
Far beyond the other cases
With no ground wire there is no shielding and little current sharing — the arrester directly clamps the conductor voltage and absorbs a very large part of the lightning energy. Even with 11 arrester towers the energy (\(750\ \text{kJ}\)) is far above the other cases.
Section 10
When Energy Duty Is Worst — and Lowest
Table 10 — Conditions giving the worst and lowest arrester energy duty.
| Worst arrester duty when… | Lowest arrester duty when… |
|---|
| No ground wire or neutral; stroke hits the phase conductor directly; high stroke current; only a few towers have arresters; high footing resistance; span reflections do not share the energy | Stroke hits the ground wire; phase is only coupled indirectly; multiple towers have arresters; current sharing is effective |
Section 12
Why Timing Is Critical
The travelling voltage wave reaches the protected tower after \(T_1\) and the unprotected tower after \(T_2\). At the protected tower the arrester only operates once the voltage reaches its discharge level, which takes:
\[ t_A = \frac{E_A}{S} \]
where \(S\) is the voltage steepness. After the arrester operates it sends a negative reflected wave back along the conductor, which can reduce the unprotected tower's voltage — but it arrives after a delay.
A race against flashover
If the unprotected tower's voltage rises too quickly, it can flash over before the arrester's relief reflection arrives. High steepness or low arrester discharge voltage make the arrester operate sooner.
Section 14
A 13.8 kV Distribution Example: Wood Helps
At lower voltage the insulation level is much lower, so nearly all direct strokes to the conductor cause flashover. The structure material changes the effective CFO — and therefore how well every-second-tower arresters help:
Table 11 — How structure material and effective CFO change every-second-tower protection.
| Structure | Effective CFO | Flashover Reduction (Arresters Every 2nd Tower) |
|---|
| Steel or concrete pole / crossarm | \(\approx 170\ \text{kV}\) (insulator only) | \(\approx 50\%\) — only the protected towers are protected |
| Wood pole / crossarm | \(\approx 1017\ \text{kV}\) | \(\approx 69\%\) — the adjacent arrester partly protects the intermediate tower |
Wood adds insulation
Wood crossarms / poles raise the effective CFO and improve the benefit of every-second-tower arresters (\(69\%\) vs \(50\%\)). Steel and concrete provide no such extra insulation.
Section 17
The Most Important Message: It Depends Where Lightning Hits
Table 13 — Ranking stroke terminations by arrester severity and the reason.
| Termination | Severity | Why |
|---|
| Stroke to ground wire | Least severe | Ground wire carries most of the surge; phase is only coupled; lower arrester current; adjacent arresters help protect unprotected towers |
| Phase stroke, with ground wire / neutral | More severe | Phase directly excited; arrester must clamp the phase voltage; coupling helps but not always enough |
| Phase stroke, no ground wire / neutral | Most severe | No shielding or coupling; arrester absorbs a large part of the surge; current and energy become very high |
Section 18
How to Read the Equations
You don't need to memorise every equation — they all serve four steps that end in one question:
Surge voltage \(e_c = \dfrac{IZ}{2}\), \(S = \dfrac{S_i Z}{2}\)
Arrester operates \(t_A = \dfrac{E_A}{S}\)
Reflections (footing, arrester, coupling)
Compare \(E_{\text{insulation}}\) with CFO
A surge current becomes a voltage through the surge impedance (halved because it travels both ways); high steepness or low \(E_A\) makes the arrester operate sooner; reflections decide whether the unprotected tower sees a dangerous voltage; and the final test is simply:
\[ E_{\text{insulation}} > \text{CFO} \;\Rightarrow\; \text{flashover}, \qquad E_{\text{insulation}} < \text{CFO} \;\Rightarrow\; \text{no flashover} \]
Section 19
Five Practical Design Lessons
Table 14 — Five practical design lessons for line arresters explained.
| Lesson | What It Means |
|---|
| 1 — Arrester current ≠ lightning current | A 100 kA stroke to the ground wire may give only \(6\ \text{kA}\) through the arrester; a 12.3 kA shielding failure may give \(9\)–\(10\ \text{kA}\). Location is critical |
| 2 — Energy depends on protected towers | More towers with arresters usually reduce per-arrester energy; hand equations may over- or under-estimate |
| 3 — Every-second-tower may not protect | For HV lines, the intermediate tower can remain essentially unprotected, especially for direct phase strokes |
| 4 — Ground wire / neutral helps | Coupling reduces insulation stress, depending on coupling factor, footing resistance, steepness, span and \(E_A\) |
| 5 — Wood adds insulation | Wood crossarms / poles raise effective CFO and improve every-second-tower arresters; steel / concrete do not |
Section 20
A Mental Picture and the Final Message
A simple mental picture
Think of a lightning stroke as a fast water hammer in a pipe system. At each tower some surge goes to earth, some reflects back, some continues forward, some couples into nearby conductors, and the arrester opens a controlled discharge path — like a pressure-relief valve. But if the surge reaches an unprotected tower before the relief effect arrives, flashover occurs. That is why distance and timing matter so much.
Equation Summary
Arrester Energy
\(\displaystyle W_A = \int v_A(t)\,i_A(t)\,dt\)
Energy is the integral of arrester voltage times current over time.
Approximate Energy Estimate
\(\displaystyle W_A \approx E_A\,I_A\,T\)
A quick estimate from discharge voltage, current and duration.
Travelling-Wave Voltage Steepness
\(\displaystyle S = \frac{S_i Z}{2}\)
Current steepness becomes voltage steepness, halved for two directions.
Arrester Operating Time
\(\displaystyle t_A = \frac{E_A}{S}\)
How soon the arrester reaches its discharge voltage.
Insulation Stress
\(\displaystyle E_{\text{ins}} = E_{\text{phase}} - E_{\text{gnd}}\)
The insulation sees the phase-to-ground-wire/neutral difference.
Flashover Criterion
\(\displaystyle E_{\text{ins}} < \text{CFO}\)
Insulation is safe while its voltage stays below the CFO.
Coupling Factor
\(\displaystyle C = \frac{Z_m}{Z_c}\)
How closely the ground wire/neutral follows the phase voltage.
Memory map. For energy: estimate crest current (travelling-wave) → estimate duration (inductance time constant) → \(W \approx E_A I_A T\) → confirm with EMTP®. For intermediate towers: find when the surge reaches each tower → when the arrester operates (\(t_A = E_A/S\)) → the negative reflected wave → the insulation voltage → compare with CFO.
Final engineering message
Line arrester performance cannot be judged by arrester rating alone — it must be studied together with line geometry, travelling-wave behaviour, footing resistance, span length, coupling, insulation CFO and lightning steepness. Arresters on every second tower may cut cost, but they do not always protect the intermediate tower: protection is far better for strokes to the ground wire than for direct strokes to the phase conductor, and on unshielded or high-voltage lines the intermediate towers may remain essentially unprotected.
