Power System Transients

Multiple-Conductor Traveling Waves and Coupling

Real lines carry coupled waves, not independent ones. Each conductor has a self-surge impedance, and neighbours are linked by a mutual surge impedance Z₁₂. From this come the equivalent surge impedance of tied ground wires, Z_e = (Z + (n−1)Z_m)/n, and the coupling factor C = Z₁₂/Z₁ that induces a voltage e₂ = C e₁ on a quiet conductor — the mechanism that lowers insulator stress against backflashover and reshapes the surge reaching the substation.

Reading time ≈ 40 min

Section 1

Why Multiple-Conductor Waves Matter

Single-conductor theory says \(e = iZ\). But multiple-conductor behaviour appears whenever two or more nearby conductors — phase conductors, ground/shield wires, parallel downleads or bundle subconductors — are close enough to be electromagnetically coupled: a voltage or current wave on one conductor induces a voltage on another through mutual surge impedance. So the voltage on conductor 1 is partly caused by the current in conductor 2, and the two-conductor relationship is coupled:

\[ e_1 = i_1 Z_1 + i_2 Z_{12} \qquad e_2 = i_1 Z_{12} + i_2 Z_2 \]

Parameter definitions

\(e_1, i_1\) — voltage and current wave on the energised conductor 1; \(e_2, i_2\) — voltage and current wave on conductor 2; \(Z_1, Z_2\) — self-surge impedances; \(Z_{12}\) — mutual surge impedance (\(Z_{12} = Z_{21}\) for a reciprocal passive geometry); \(Z_m\) — mutual impedance when all mutuals are taken equal (\(Z_m = Z_{12}\) for two conductors); \(Z\) — common self-impedance of identical conductors; \(Z_e\) — equivalent impedance of a tied group; \(n\) — number of identical tied conductors; \(C\) — coupling factor. A full table is given in the summary (Section 16).

Section 2

Self-Surge Impedance

\(Z_1\) and \(Z_2\) are the self-surge impedances — the surge impedance each conductor would have if the other did not exist. More precisely, \(Z_1\) is the voltage-to-current ratio for a wave on conductor 1 once conductor 2’s influence is carried by the coupled equations:

\[ Z_1 = 60\,\ln\!\frac{2h_1}{r_1}, \qquad Z_2 = 60\,\ln\!\frac{2h_2}{r_2} \]

If the conductors differ in height or radius, \(Z_1 \neq Z_2\).

Section 3

Mutual Surge Impedance

The mutual surge impedance \(Z_{12}\) represents the coupling between the conductors — how much voltage appears on one conductor due to current traveling on the other. It exists because the electric and magnetic fields around one conductor link with nearby conductors, so a fast current wave on one produces an induced voltage wave on the other. The term \(i_2 Z_{12}\) is the voltage induced on conductor 1 by the current on conductor 2, and \(i_1 Z_{12}\) is the voltage on conductor 2 from the current on conductor 1. The coupling is symmetric (\(Z_{12} = Z_{21}\) for a reciprocal passive geometry): the same \(Z_{12}\) appears in both equations. When all mutual terms are taken equal we write \(Z_m\); for two conductors \(Z_m = Z_{12}\).

Section 4

Matrix Form and Solving for Currents

The pair is exactly an impedance-matrix relation — the same idea as in steady-state studies, but for traveling waves. The diagonal terms are the self-surge impedances and the off-diagonal terms the mutual surge impedances, so each conductor’s voltage is affected by its own current wave and by current waves on nearby conductors:

\[ \begin{bmatrix} e_1 \\ e_2 \end{bmatrix} = \begin{bmatrix} Z_1 & Z_{12} \\ Z_{12} & Z_2 \end{bmatrix} \begin{bmatrix} i_1 \\ i_2 \end{bmatrix} \]

When the voltages are known and the currents are needed, invert the matrix. With determinant \(\Delta = Z_1 Z_2 - Z_{12}^2\):

\[ i_1 = \frac{Z_2\,e_1 - Z_{12}\,e_2}{Z_1 Z_2 - Z_{12}^2}, \qquad i_2 = \frac{Z_1\,e_2 - Z_{12}\,e_1}{Z_1 Z_2 - Z_{12}^2} \]

Because \(Z_{12}\) couples the conductors, the current in one conductor depends on both voltages.

Section 5

Equivalent Surge Impedance

Tied vs coupled conductors

Two distinct situations occur. In tied conductors the conductors are electrically connected and share the injected current (e.g. two ground wires bonded at the tower) — this section. In coupled conductors (Sections 8–11) they are not connected, but a traveling wave on one induces a voltage on the other.

Often several conductors are tied together or rise to the same traveling voltage — for example the two overhead ground wires on a tower struck by lightning. We then want one combined surge impedance \(Z_e\), which is central to computing the tower-top voltage. Take two conductors with \(Z_1 = Z_2 = Z\), mutual \(Z_{12} = Z_m\), and the same voltage \(e_1 = e_2 = e\). By symmetry \(i_1 = i_2\), the total current is \(i_T = 2i_1\), and \(Z_e = e/i_T\):

\[ Z_e = \frac{e}{i_T} = \frac{Z + Z_m}{2} \]

For two identical tied conductors the equivalent impedance reduces to \(Z_e = (Z+Z_m)/2\); because \(Z_m\) is positive, this is higher than \(Z/2\).

Section 6

Coupling Raises the Equivalent Impedance

Without coupling, two identical conductors in parallel would give \(Z_e = Z/2\). With coupling \(Z_e = (Z + Z_m)/2\), and since \(Z_m > 0\) the equivalent impedance is higher than simple parallel impedance. Two overhead ground wires therefore do not behave like two independent surge impedances in parallel — their electromagnetic coupling makes \(Z_e\) larger.

Section 7

Generalising to n Conductors

For \(n\) identical tied conductors, all with self-impedance \(Z\) and equal mutual impedance \(Z_m\) between every pair:

\[ Z_e = \frac{Z + (n-1)\,Z_m}{n} \]

\(n\): number of identical tied conductors
\(Z\) / \(Z_m\): common self / mutual surge impedance
\(Z_e\): equivalent surge impedance of the tied group

Without coupling, \(n\) identical parallel conductors would give \(Z/n\). With coupling, the mutual terms raise the equivalent voltage for a given total current, so \(Z_e = [Z+(n-1)Z_m]/n\) — not simply \(Z/n\). (Notation: this section uses \(Z, Z_m\) for identical conductors; the two-conductor coupling sections use \(Z_1, Z_{12}\).)

Table 1 — Equivalent surge impedance for equal coupled conductors.

Conductors	\(Z_e\)
2	\((Z + Z_m)/2\)
3	\((Z + 2Z_m)/3\)
\(n\)	\((Z + (n-1)Z_m)/n\)

Unequal conductors

If the self- and mutual impedances are not all equal, the same formula stays accurate within about 2% if \(Z\) is taken as the average self-surge impedance and \(Z_m\) as the average mutual surge impedance — convenient for practical work.

Section 8

The Coupling Factor

Suppose a wave travels on conductor 1 while conductor 2 is open, so \(i_2 = 0\) — a “quiet” conductor, meaning one not directly struck or energised by the incident surge but receiving an induced wave through coupling. With \(e_1 = i_1 Z_1\) and \(e_2 = i_1 Z_{12}\), the coupling factor \(C\) — the fraction of the energised-conductor voltage that appears on the neighbour under this simplified traveling-wave coupling assumption — is:

\[ C = \frac{e_2}{e_1} = \frac{Z_{12}}{Z_1} \qquad\Longrightarrow\qquad e_2 = C\,e_1 \]

\(Z_{12}\): mutual surge impedance between conductors 1 and 2
\(Z_1\): self-surge impedance of the energised conductor

Worked example

If \(Z_1 = 450\) Ω and \(Z_{12} = 135\) Ω, then \(C = 135/450 = 0.30\). A 1000 kV ground-wire surge induces \(\approx 300\) kV on the phase conductor, reducing the approximate insulator stress to \(1000 - 300 = 700\) kV. The coupling factor is geometry-dependent — it changes with conductor spacing, height, radius or bundle radius, earth-return assumptions, and corona (see Part Four) — so it is not a universal constant.

Section 9

What the Coupling Factor Means

\(C\) is the fraction of a traveling voltage on one conductor that appears on another. If \(C = 0.2\) and \(e_1 = 1000\) kV, then conductor 2 rises to \(e_2 = 0.2 \times 1000 = 200\) kV — even with no direct source applied to it. This induced rise is central to lightning analysis.

Section 10

Why Coupling Helps Against Backflashover

In a tower-stroke case the ground wire (shield wire) rises in voltage first. The phase conductor is not directly struck, but it is pulled upward by coupling, so the stress across the insulator string is the difference between the two:

\[ V_{\text{phase}} \approx C\,V_{gw} \qquad V_{\text{ins}} \approx V_{gw} - V_{\text{phase}} \approx (1 - C)\,V_{gw} \]

\(V_{gw}\): ground-wire / tower-side voltage
\(V_{\text{phase}}\): induced phase-conductor voltage
\(V_{\text{ins}}\): voltage across the insulator string

Higher coupling → lower insulator stress

For a typical tower or shield-wire stroke the induced phase voltage has the same polarity as the ground-wire voltage (here \(C\) is treated as a positive coupling factor), so the difference \((1-C)V_{gw}\) is reduced and backflashover risk falls. The sign of the induced voltage depends on the chosen voltage reference and current direction, so this benefit assumes the same-polarity, tower-stroke condition. (For induced surges on otherwise quiet lines, higher coupling can instead mean a higher conductor surge.)

Note the balance: the same coupling that reduces local insulator stress also transfers part of the surge onto the phase conductor, and that induced surge can then travel toward the substation and add to equipment stress.

Section 11

Two Ground Wires and One Phase Conductor

The practical shielding case: two ground wires raised to voltage \(e_g\) by a stroke, with the phase conductor carrying no initial current (\(i_c = 0\)) but acquiring an induced voltage. The coupling factor uses the equivalent (average) mutual surge impedance between the ground-wire group and the phase conductor, \(Z_{mc}\), over the combined ground-wire surge impedance \(Z_e\):

\[ C = \frac{Z_{mc}}{Z_e} \qquad\Longrightarrow\qquad e_c = C\,e_g \]

Bringing the ground wires closer to the phase conductor raises \(Z_{mc}\) and hence \(C\), coupling more voltage onto the phase — which is helpful for backflashover (the phase rises with the tower) even though it raises the absolute conductor surge.

Section 12

Multiple Conductors at a Discontinuity

The boundary rules \(e'' = e + e'\) and \(i'' = i - i'\) must now be written for every conductor, with the coupled \(e = i_1 Z_1 + i_2 Z_{12}\) relations holding for the incoming, reflected and transmitted sets. The consequence: a surge arriving on just the ground wire produces reflected waves on both the ground wire and the phase conductor — reflection is no longer independent per conductor but a coupled traveling-wave problem.

Example (surge on the top conductor only): injecting voltage and current into the top conductor with \(i_2 = 0\) produces a voltage on the bottom conductor exactly equal to \(e_2 = C\,e_1\) — confirming the coupling-factor result.

Section 13

A Surge Approaching a Tower

The key practical case: a ground-wire wave \(e_1\) and a phase-conductor wave \(e_2\) approach a tower grounded through footing resistance \(R\). The ground wire connects to the tower (so its voltage there is set by the footing current, \(e_1'' = i_R R\)), while the phase conductor passes the tower insulated but remains coupled to the ground wire. Each tower therefore acts as a shunt grounding point for the ground wire:

part of the ground-wire surge current drains into the footing resistance;
part continues along the ground wire;
the ground-wire voltage is reduced, and the phase voltage is modified through coupling.

So each tower gradually attenuates the ground-wire surge as the wave travels toward the substation.

Section 14

Many Towers and the Limiting Result

The outgoing waves after tower \(n\) become the incoming waves at tower \(n+1\), computed by the same equations. As \(n \to \infty\) the phase-to-ground-wire relationship approaches a limit governed by the coupling factor:

\[ n \to \infty \;\Longrightarrow\; e_2 \to C\,e_1 \]

In this limit \(e_1\) is the local ground-wire voltage at the tower/location considered (after attenuation through previous towers), not necessarily the original stroke voltage at the struck tower. After many towers, the coupled phase voltage is governed by \(C\). If lightning causes a flashover from tower/ground wire to the phase conductor, the arc ties them together and the voltages become equal:

\[ \text{after flashover:}\quad e_1 = e_2 \]

That injects a surge onto the phase conductor which then travels toward the substation — the limiting relationship above is later used to estimate the crest of the backflashover surge arriving at the substation entrance.

Section 15

Substation Entrance Surges and EMTP®

A backflashover suddenly raises the phase conductor to the tower/ground-wire potential, launching traveling waves toward the substation. The crest arriving at the entrance depends on the tower voltage at flashover, the ground-wire/phase coupling, attenuation through successive towers, reflections, footing resistances, line configuration and distance to the station — so the entrance surge is not simply the original tower voltage; it is reshaped by the multi-conductor system.

EMTP® solves this with multi-phase distributed or frequency-dependent line models (JMarti / wideband) using full impedance and admittance matrices with modal transformation, plus tower surge impedance and footing models, flashover switches and station-entrance arresters. The scalar coupling factor \(C\) is excellent for hand calculations, trend-checking and interpretation, but the detailed wave propagation among coupled conductors is solved with these full matrix / modal models. Even so, \(C\) — which sets the voltage across the line insulation — is why coupling matters so much for backflashover rate, shielding-failure analysis, footing-resistance effects, arrester placement and lightning insulation coordination.

Section 16

Memory Map and Summary

Table 2 — Symbol reference.

Symbol	Meaning
\(Z_1\) / \(Z_2\)	Self-surge impedance of conductor 1 / 2
\(Z_{12}\)	Mutual surge impedance between conductors 1 and 2
\(Z\) / \(Z_m\)	Self / mutual surge impedance of identical conductors
\(Z_e\) / \(n\)	Equivalent impedance of a tied group / number of tied conductors
\(e_1\) / \(e_2\)	Voltage wave on energised / neighbouring conductor
\(i_1\) / \(i_2\)	Current waves on conductors 1 and 2
\(C\)	Coupling factor (\(Z_{12}/Z_1\) or \(Z_{mc}/Z_e\))
\(V_{gw}\) / \(V_{\text{phase}}\)	Ground-wire/tower-side voltage / induced phase-conductor voltage
\(V_{\text{ins}}\)	Insulator voltage stress, \(\approx (1-C)V_{gw}\)
\(R\)	Tower footing resistance (surge at a tower)

Equation Summary

Conductor 1 voltage

\(\displaystyle e_1 = i_1 Z_1 + i_2 Z_{12}\)

Conductor 2 voltage

\(\displaystyle e_2 = i_1 Z_{12} + i_2 Z_2\)

Equivalent surge impedance

\(\displaystyle Z_e = \frac{Z + (n-1)Z_m}{n}\)

Coupling factor

\(\displaystyle C = \frac{Z_{12}}{Z_1} = \frac{Z_{mc}}{Z_e}\)

Induced voltage

\(\displaystyle e_2 = C\,e_1\)

After flashover

\(\displaystyle e_1 = e_2\)

Key messages

On real lines, waves are coupled: each conductor has a self-impedance \(Z\) and is linked to others through a mutual impedance \(Z_{12}\).
Tied conductors have equivalent impedance \(Z_e = (Z + (n-1)Z_m)/n\) — higher than simple parallel because coupling adds \(Z_m\).
The coupling factor \(C = Z_{12}/Z_1\) induces a voltage \(e_2 = C\,e_1\) on a neighbouring conductor with no current of its own.
Higher ground-wire/phase coupling reduces the insulator stress \(V_{\text{tower}} - V_{\text{phase}}\) — it lowers backflashover risk.
Each tower grounds the shield wire and attenuates the surge; after many towers the phase voltage tends to \(C\,e_1\), and after flashover \(e_1 = e_2\).
The substation-entrance backflashover surge is reshaped by the coupled multi-conductor system — the physics behind EMTP® multi-phase line models.

Part Three Reading now

Multiple-Conductor Traveling Waves and Coupling

Self and mutual surge impedance, equivalent surge impedance Z_e of tied ground wires, and the coupling factor C = Z₁₂/Z₁ that lowers insulator stress and shapes backflashover surges.

Series progress 3 of 4

Part Four

Tower Surge Impedance and Corona

Tower surge impedance as a practical field-theory equivalent, and how corona adds capacitance on the rising front — lowering surge impedance and raising the coupling factor.

Previous Lattice Diagrams and the Stroke to a Tower

Tracking repeated reflections in time — a finite cable on a line, and a lightning stroke to a tower with its footing and inductive voltage components.