Power System Transients

Lattice Diagrams and the Stroke to a Tower

When a network has many discontinuities, the lattice (time-position) diagram tracks every reflection and tells you when each peak arrives. Two applications: a finite cable on an overhead line, where surge impedance controls the early response and the termination the final one; and a lightning stroke to a tower, where the tower-top voltage combines a footing component I(R_i ∥ Z_g/2) with a tower inductive component L_T dI/dt — reduced by adjacent-tower reflections (K_sp) — the traveling-wave basis of backflashover.

Reading time ≈ 40 min

Section 1

Why a Lattice Diagram Is Needed

At a single discontinuity the reflected and transmitted waves follow directly from the reflection and transmission coefficients \(\Gamma_v = (Z_k-Z)/(Z_k+Z)\) and \(T_v = 2Z_k/(Z+Z_k)\), where \(\Gamma_v\) is the voltage reflection coefficient, \(T_v\) the initial voltage transmission coefficient, \(Z\) the surge impedance of the incoming conductor, and \(Z_k\) the impedance seen beyond the discontinuity. But real networks have many discontinuities — line→cable→transformer, or tower→ground wire→adjacent towers. Each one launches new reflected and transmitted waves, which travel back and forth and spawn still more waves.

The question becomes: how do you track all the reflections in time? The answer is the lattice diagram.

Section 2

What a Lattice Diagram Is

A lattice diagram is a time-position diagram: the horizontal axis is position along the conductor system and the vertical axis is time (time increases downward). A sloping line represents a traveling wave, and its slope is set by the propagation velocity — a slower cable wave draws a steeper line than an overhead-line wave. Every time a wave meets a discontinuity the reflection and transmission coefficients are applied, so each wave carries a running product of coefficients.

For example, a wave first transmitted through a junction with coefficient \(T_1\) and then reflected at the remote end with coefficient \(\Gamma_2\) returns with magnitude \(e\,T_1\,\Gamma_2\). The diagram shows where each wave goes, when it arrives, how it reflects and transmits, and what voltage it adds at each point.

Not replaced by EMTP® — it explains EMTP®

EMTP® solve these problems numerically, but the lattice diagram remains the engineer’s interpretation tool: it tells you which reflection causes a peak, when the peak should occur, why a voltage rises after a delay, why a cable remote end can double (under an open or high-impedance termination) — and whether a simulation result is physically reasonable.

Section 3

A Finite Cable on an Overhead Line

For an infinite cable the junction voltage is computed once. For a finite cable the surge travels to the remote end, reflects, and returns to the line–cable junction — so the junction voltage changes with time: the initial entrance voltage is not the later entrance voltage. Consider an overhead line feeding a finite cable terminated in a resistor \(R\) (the resistance connected at the remote end of the cable — a general termination here, often a simplified transformer/load representation):

The surge reaches the line–cable junction at \(t = 0\); part enters the cable.
It travels the cable and meets \(R\) at the far end; part reflects back.
The reflection returns to the junction, where it splits again — setting up repeated reflections between the junction and the cable end.

For a cable of length \(\ell_c\) and wave velocity \(v_c\), the one-way travel time is \(T_c = \ell_c/v_c\), and the first reflection from the cable end returns to the junction after \(2T_c\) — the basis of every arrival time on this page.

Section 4

Building the Diagram

The construction is mechanical. Start at \(t = 0\) when the surge reaches the junction; draw the transmitted wave diagonally to the cable end; multiply by the cable-end reflection coefficient; draw the reflected wave back to the junction; split it into new transmitted and reflected waves; repeat. The voltage at any point is then the sum of all waves arriving there at their arrival times:

The one rule of the lattice diagram

Voltage at a point = the sum of every arriving wave contribution at that instant, each delayed by its own travel time. New wave = old wave × coefficient.

Section 5

Travel Time and the Delayed-Wave Sum

Using the one-way cable travel time \(T_c = \ell_c/v_c\), a round trip to the end and back is \(2T_c\). So remote-end reflections return to the junction at \(2T_c, 4T_c, 6T_c, \dots\), and the junction voltage is a sum of progressively delayed copies of the incident surge:

\[ T_c = \frac{\ell_c}{v_c} \] \[ e_T(t) = \beta\,e(t) + \delta\alpha\beta\,e(t - 2T_c) + \delta\alpha\phi\alpha\beta\,e(t - 4T_c) + \cdots \]

\(\beta,\alpha,\delta,\phi\): junction / cable-end reflection & transmission coefficients
\(e(t-2nT_c)\): the incident surge shape, zero until \(t = 2nT_c\), then delayed

Each reflected term adds only after it has physically traveled to the end and back — that time delay is the whole point. (Some terms can be written two equivalent ways, e.g. \(\beta e = (1+\gamma)e\), simply because the junction voltage is incident plus reflected; it is not a contradiction.)

Section 6

A Linear-Front Example, and the “Wiped-Out” Cable

Take a surge with a linear front, infinite tail, time-to-crest \(t_f = 4\) µs, and cable travel time \(T_c = 1\) µs. Reflections return at \(2T_c, 4T_c, 6T_c = 2, 4, 6\) µs — and because the surge is still rising before 4 µs, the reflections arriving in that window reshape the crest. The maximum occurs at \(t = 4\) µs and reaches about \(0.075\,e\).

Early time vs late time

As \(t \to \infty\) the repeated reflections effectively “wipe out” the cable: the response becomes that of a surge on the overhead line terminated directly in \(R\). So the initial response is controlled by the cable surge impedance, while the final response is controlled by the actual termination \(R\) — the reflections carry the system from one to the other.

Section 7

Why This Matters for Cable Studies

For cable-connected equipment the first peak may not be at the instant the surge enters the cable. Important peaks can arrive later, from reflection off an open cable end, a transformer terminal, an arrester, a GIS transition, or sheath/bonding discontinuities. So EMTP® voltage peaks must be checked at the sending end, receiving end, line–cable junction, transformer terminal, arrester terminal and cable sheath/bonding points — the lattice diagram explains why.

Section 8

A Lightning Stroke to a Tower

The same reflection-tracking idea now applies to a lightning stroke to a tower, where waves travel down the tower, into the footing, and out along the ground wires toward adjacent towers — a special application of the lattice method. A stroke of current \(I\) to the tower top produces a voltage \(e\) there, which launches waves down the tower and outward along the ground wires in both directions. The initial tower-top voltage is set by the stroke current and the effective impedance it sees:

Backflashover vs shielding failure

Backflashover occurs when the tower or ground-wire voltage rises so high that the insulator string flashes over from the tower/crossarm side to the phase conductor. It is distinct from a shielding failure, where the stroke terminates directly on the phase conductor. This page (stroke to the tower or shield wire, phase conductor flashed indirectly) is about backflashover.

\[ e = I \times Z_{eq} \]

\(Z_{eq}\): combined impedance of tower and ground wires at the tower top

Section 9

Tower Impedance and the Z_g/2 Trick

The tower top sees the tower surge impedance \(Z_T\) downward and the ground-wire surge impedance \(Z_g\) outward in two directions. The two ground-wire directions are in parallel, so the equivalent ground-wire impedance is \(Z_g/2\). A common simplifying assumption keeps the algebra clean:

\[ Z_T \approx \frac{Z_g}{2} \]

This is why the tower is compared against \(Z_g/2\) rather than \(Z_g\): the struck tower drains into both ground-wire spans at once.

Section 10

Tower Travel Time and the Three Voltages

A wave down a tower of height \(h\) takes \(T_T = h/v_T\) — small (e.g. \(30\) m at \(300\) m/µs gives \(0.1\) µs) but not zero for a steep lightning front. The analysis defines three tower-top voltages:

Table 1 — The three tower-top voltages.

Symbol	Meaning
\(V_T\)	Initial tower-top voltage before footing reflections return — the tower inductive component
\(V_{TT}\)	Crest voltage at the tower top, including the first important reflection from the footing
\(V_F\)	Final voltage once the current distribution reaches its steady traveling-wave limit

Section 11

The Tower Inductive Voltage

The tower does not behave as a pure resistance. For a steep front the tower inductance produces a real voltage:

\[ V_T = L_T\,\frac{dI}{dt}, \qquad L_T = Z_T\,T_T \;\Rightarrow\; V_T = Z_T\,T_T\,\frac{I}{t_f} \]

\(V_T\): tower inductive voltage component
\(L_T\): equivalent tower inductance (\(= Z_T T_T\))
\(Z_T\): tower surge impedance
\(T_T\): tower travel time, top to footing
\(I\): lightning stroke current crest
\(t_f\): current front time (\(dI/dt \approx I/t_f\), a linear-front approximation)

Since \(1\,\Omega \cdot 1\,\text{s} = 1\,\text{H}\), the product \(Z_T T_T\) behaves as an equivalent tower inductance. So even with low footing resistance, a fast current front can drive a high tower-top voltage: the inductive component grows with higher tower surge impedance, greater tower height/travel time, higher current crest, and shorter front time \(t_f\). The relation \(dI/dt \approx I/t_f\) is exact only for a linear current front, not for every wave shape.

Section 12

The Final Footing Voltage and Backflashover

The final voltage is governed by the footing resistance in parallel with the ground-wire equivalent:

\[ \frac{V_F}{I} \approx R_i \parallel \frac{Z_g}{2} \qquad\xrightarrow{\;Z_g \gg R_i\;}\qquad V_F \approx I R_i \]

\(V_F\): final / late-time tower footing voltage
\(I\): lightning current entering the tower/ground-wire system
\(R_i\): effective footing resistance of the struck tower
\(Z_g\): surge impedance of one ground-wire path
\(Z_g/2\): two ground-wire directions in parallel at the struck tower

The factor \(Z_g/2\) appears because the ground-wire current can launch waves in two directions away from the struck tower, so the two ground-wire paths appear in parallel there. This is the familiar footing-resistance rise of the tower as current enters the earth. It matters because a high tower potential drives the stress across the line insulation: if \(V_{\text{tower}} - V_{\text{phase}}\) gets large enough, the gap flashes from tower to conductor — a backflashover. It also raises step and touch voltages near the tower.

Section 13

Worked Example — Stroke to Tower

With \(Z_g = 350\) Ω, \(Z_T = 200\) Ω, \(R_i = 20\) Ω, \(h = 30\) m, \(t_f = 2\) µs:

Table 2 — Stroke-to-tower results (per unit of stroke current).

Quantity	Value	Meaning
\(V_{TT}/I\)	26.13 Ω	Tower-top crest voltage
\(V_F/I = V_R/I\)	17.95 Ω	Final / footing-resistance component
\(I_R/I\)	0.898	89.8% of stroke current flows through the footing

So only ~10% leaves along the ground wires. The footing component dominates the crest (~78% by the full decomposition), with the tower traveling-wave/inductive component making up the remaining ~22% — not negligible. The approximations assume \(Z_T = Z_g/2\); here \(Z_T = 200\) vs \(Z_g/2 = 175\) Ω, giving ~5% error for \(t_f > 1\) µs — acceptable given the large uncertainty in tower impedance, footing resistance and lightning waveform. The same equations give the voltage at any point A on the tower by using the travel time \(T_A\) from the top down to A (needed later for the voltage across the insulator string at the crossarm).

Section 14

The Effect of Adjacent Towers

The single-tower result assumed infinitely long ground wires. In reality the ground wire is supported by adjacent towers; a wave reaching the next tower partly diverts into that tower’s footing and partly reflects back, and the returning reflection reduces the struck-tower voltage — extra grounding, but only after the travel delay. A folded-line representation collapses the two ground-wire directions into one equivalent line of impedance \(Z_g/2\).

The two footings are kept distinct: \(R_i\) is the effective footing resistance of the struck tower under high lightning current (high-current soil ionisation may lower it), while \(R_0\) is the footing resistance of adjacent towers, taken closer to the measured low-current value because their current is much smaller. The span travel time is \(T_s = (\text{span})/v\) (e.g. 300 m at 300 m/µs = 1 µs), so reflections return after \(2T_s\). A span-reflection correction factor \(K_{sp}\) — less than 1 when adjacent-tower reflections reduce the crest — scales it:

\[ V_{TT,\text{corr}} = K_{sp}\,V_{TT}, \qquad\text{include terms only where } 1 - \frac{nT_s}{t_f} > 0 \]

\(K_{sp}\): span-reflection correction factor — the crest reduction from adjacent-tower reflections arriving before crest (\(K_{sp} < 1\))
\(n\): reflection index in units of \(T_s\); the reflection arrives at \(nT_s\)

The condition \(1 - nT_s/t_f > 0\) means only reflections arriving before the current front reaches its crest (\(t_f\)) are included in the crest correction; a reflection arriving after \(t_f\) does not reduce the crest — it mainly affects the tail. Here \(nT_s\) is the actual arrival time of a reflected component; for the first reflection from an adjacent tower this is typically \(2T_s\). Example: \(t_f = 6\) µs, \(T_s = 1\) µs, \(Z_g = 300\), \(Z_T = 150\), \(R_0 = 40\), \(R_i = 20\) Ω gives \(K_{sp} = 0.8388\), corresponding to an approximate 16% reduction \((1 - 0.8388 = 0.161)\) in the struck-tower crest. Reflections from the second tower out lower it by less than 1%, so only the first adjacent towers usually matter.

Section 15

Tail Decay and Adjacent-Tower Current

Even when adjacent reflections do little to the crest, they decay the tail: current spreads into the ground wires and adjacent footings, so the idealised infinite tail falls away. Beyond a start time \(t_0 \approx t_f + 2T_T\) the tail is approximated by a single exponential, measured from \(t_0\) (not from \(t = 0\)):

\[ e_v(t) = V_F\,e^{-(t - t_0)/\tau}, \qquad t_0 \approx t_f + 2T_T, \qquad \tau \sim \frac{Z_g\,T_s}{R_i}\ \ (L = Z_g T_s) \]

\(e_v(t)\): late-time voltage tail approximation
\(\tau\): approximate tail-decay time constant
\(t_0\): start of the tail-decay approximation

The \(\tau \sim Z_g T_s/R_i\) form is a simplified equivalent \(L/R\) estimate for the late-time ground-wire/footing decay (\(L = Z_g T_s\)) — not an exact universal expression. Larger ground-wire impedance and longer spans lengthen the tail; footing resistances set the decay path (typical \(R_0/R_i = 2\)–5). In many practical cases the adjacent-tower footing current is only a small fraction — often of the order of 4–8% of the struck-tower footing current, depending on span length, ground-wire impedance, footing resistance and stroke-front time — which is why adjacent towers see little ionisation and their \(R_0\) can be held at the measured low-current value, while the struck tower’s \(R_i\) may be reduced by high current.

Section 16

Memory Map and Summary

Table 3 — Symbol reference.

Symbol	Meaning
\(\Gamma_v\) / \(T_v\)	Voltage reflection / transmission coefficient
\(Z\) / \(Z_k\)	Surge impedance of incoming line / impedance beyond a discontinuity
\(R\)	Remote-end terminating resistance of the finite cable
\(T_c\)	One-way cable travel time (\(\ell_c/v_c\))
\(V_T\) / \(L_T\)	Tower inductive voltage / equivalent tower inductance
\(Z_T\) / \(T_T\)	Tower surge impedance / tower travel time
\(I\) / \(t_f\)	Lightning stroke current crest / current front time
\(V_F\)	Final / late-time footing voltage
\(R_i\) / \(R_0\)	Footing resistance of struck tower / adjacent tower
\(Z_g\) / \(T_s\)	Ground-wire surge impedance / one-way span travel time
\(K_{sp}\)	Adjacent-span reflection correction factor (\(< 1\))
\(V_{TT}\) / \(V_{TT,\text{corr}}\)	Tower-top crest before / after adjacent-tower correction
\(\tau\) / \(e_v(t)\)	Approximate tail-decay time constant / late-time tail voltage

Equation Summary

Cable travel time

\(\displaystyle T_c = \frac{\ell_c}{v_c}\)

Tower inductive voltage

\(\displaystyle V_T = L_T\,\frac{dI}{dt}\)

Tower inductance

\(\displaystyle L_T = Z_T\,T_T\)

Final footing voltage

\(\displaystyle V_F \approx I\Bigl(R_i \parallel \tfrac{Z_g}{2}\Bigr) \xrightarrow{Z_g \gg R_i} I R_i\)

Adjacent-tower correction

\(\displaystyle V_{TT,\mathrm{corr}} = K_{sp}\,V_{TT}\)

Tail time constant

\(\displaystyle \tau \sim \frac{Z_g\,T_s}{R_i}\)

Adjacent footing current

\(\displaystyle I_{RA} \approx 4\text{–}8\%\ I_R\)

Key messages

The lattice diagram tracks repeated reflections in time (time downward); the voltage at a point is the sum of all arriving wave contributions, each delayed by its travel time.
A finite cable makes the junction voltage time-dependent: surge impedance controls the early response, the termination controls the final — reflections “wipe out” the cable as \(t \to \infty\).
For a stroke to a tower, the tower-top voltage has a footing component \(I(R_i \parallel Z_g/2)\) and a tower inductive component \(L_T\,dI/dt\) — both matter, the inductive part growing with steeper fronts.
In the example, ~90% of the stroke current flows through the struck footing; the footing dominates the crest (~78%), tower ~22%.
Adjacent towers reduce the struck-tower crest (~16% via \(K_{sp}\) in the example) and shorten the tail; their footing current is only 4–8%, so \(R_0\) stays at its measured value.
This is the traveling-wave basis of backflashover analysis — the tower voltage that EMTP® computes from tower, footing, ground-wire and adjacent-tower models.

Use for screening and interpretation

These analytical relationships are useful for checking trends and interpreting results, but final lightning insulation coordination should be based on an EMT model using appropriate line/cable models, tower-footing representation, flashover criteria and arrester models.

Part Two Reading now