Power System Transients

Tower Surge Impedance and Corona

Two practical refinements for lightning studies, taken here as travelling-wave interpretation rather than full modelling. Tower surge impedance is a field-theory equivalent, not a fixed resistance: for many conventional line studies simple cylinder or cone equivalents are adequate for screening, but a detailed tower model can matter when tower height, crossarm-level voltage, multistorey response or backflashover timing controls the result. Corona, by contrast, strongly reshapes a travelling wave: it adds capacitance on the rising front (C_n + ΔC), lowering velocity and surge impedance, pushing back the wavefront, cutting steepness and crest, and raising the coupling factor that protects the insulator string.

Reading time ≈ 40 min

Section 1

Why Tower Surge Impedance Is Difficult

For a round overhead conductor the surge impedance is straightforward, \(Z = 60\,\ln(2h/r)\). (This is an approximate expression for an isolated round conductor above a reference ground plane, used here only to show why tower surge impedance is less direct — a tower is not a uniform round conductor with a single height and radius.) A tower is not so simple: it has legs, crossarms and bracing, a width that changes with height, connections to earth and to the ground wires, and several possible voltage-reference paths. So “tower voltage” and “tower surge impedance” need careful interpretation — they are not as clean as for a single conductor.

Tower surge impedance should be treated as an engineering equivalent used to estimate the tower-top or crossarm voltage response during a fast lightning surge. It is not a fixed physical resistance and is less uniquely defined than the surge impedance of a uniform conductor above ground — which is why several approximate or empirical formulas exist.

Parameter definitions

\(Z_T\) — equivalent tower surge impedance; \(Z_e\) — equivalent impedance of two tied downleads/current paths; \(Z_m\) — mutual surge impedance; \(h\) — effective tower (or conductor) height; \(r\) — equivalent radius; \(c\) — speed of light in free space; \(Z_0\) / \(Z_c\) — surge impedance before / during corona; \(C_n\) — normal conductor capacitance; \(\Delta C\) — added corona capacitance; \(k_0\) / \(k_c\) — coupling factor before / during corona; \(V_{\text{ins}}\) — insulator voltage. A full table is given in the summary (Section 16).

Section 2

Why Tower Voltage Is Not Uniquely Defined

Voltage is the line integral of the electric field:

\[ V = \int \mathbf{E} \cdot d\mathbf{l} \]

During a very fast transient the voltage associated with a tower or conductor is not as uniquely defined as a low-frequency node voltage. Because the field is changing rapidly, the calculated or measured value depends on the reference conductor, the return path and the field distribution assumed in the model, so different but equally reasonable definitions can give different “tower voltages”. At power frequency this rarely matters; for a fast travelling wave it does — which is exactly why tower surge impedance should be treated as a practical equivalent rather than an exact physical resistance.

Section 3

The Wave Has Not Yet “Found” Ground

When a conductor at height \(h\) is suddenly energised, it does not immediately “know” the ground is there — the field must travel to ground and back, taking about

\[ t = \frac{2h}{c} \]

\(h\): conductor (or effective tower) height above ground
\(c\): speed of light in free space

At the very beginning of a steep surge the electromagnetic field has not yet fully interacted with the ground-return boundary, so the apparent surge impedance during the first moments can differ from the later established conductor-to-ground value. Only after about this time is the usual value fully established — the surge impedance is effectively time-varying at the start, and the same applies to a tower.

Section 4

The Quantity That Actually Matters

When lightning hits the tower top, current and charge waves travel down the tower and out along the ground wires, creating electric fields. If you integrate the field along the tower steel and assume it perfectly conducting, the voltage drop is zero — yet a lightning stroke clearly stresses the insulator string. So the useful quantity is not abstract “tower potential” but the:

The useful quantity

Voltage between the tower / crossarm and the phase conductor — that is where the tower surge impedance enters, and where backflashover is decided.

\[ V_{\text{ins}} = V_{\text{tower/crossarm}} - V_{\text{phase}} \]

This links directly to the coupling page: the phase conductor is pulled up by coupling, which reduces \(V_{\text{ins}}\).

Section 5

The Tower Surge-Impedance Equation

Approximating the tower as a cylinder of radius \(r\) and height \(h\), the maximum tower surge impedance has the familiar form:

\[ Z_T = 60\,\ln\!\frac{2h}{r} \]

\(Z_T\): equivalent tower surge impedance
\(h\): effective tower height
\(r\): equivalent radius of the tower representation

Because \(Z_T\) is time-varying, Sargent and Darveniza suggested using an average over the interval \(0 \le t \le 2h/c\), with one expression for a cylindrical tower and another for a conical tower. The cylinder/cone form is a simplified equivalent — useful for screening and interpretation, but the actual tower response depends on tower geometry, crossarms, bracing, ground wires, footing impedance and the current front time.

How sensitive is the result?

Backflashover estimates are usually less sensitive to the exact tower surge impedance than to footing resistance, stroke-current steepness, span reflections and ground-wire-to-phase coupling — so moderate errors in \(Z_T\) barely move the result. Even so, \(Z_T\) should still be represented with a reasonable engineering value; it depends on tower geometry, height, crossarm arrangement, equivalent radius, single vs double circuit, field-path assumptions and current front time.

Section 6

Cylinders, Cones and H-Frame Downleads

A single vertical cylinder represents a simple tower; a cone represents a wider one (double-circuit, broad lattice, wide base / narrow top) by accounting for the changing radius with height. For a wood-pole H-frame with two downleads, the surge current splits between two leads, each with self-impedance \(Z_T\) and a mutual \(Z_m\). For two identical downleads in parallel the equivalent is not simply \(Z_T/2\), because the two are electromagnetically coupled:

\[ Z_e = \frac{Z_T + Z_m}{2} \]

\(Z_e\): equivalent surge impedance of two tied downleads
\(Z_m\): mutual surge impedance between the downleads

What “two downleads” means

Here “two downleads” means two electrically bonded downward current paths represented as equivalent conductors. For a lattice tower these may represent simplified tower-leg or tower-body current paths rather than separate external down conductors. Because \(Z_m > 0\), the mutual coupling keeps \(Z_e\) above \(Z_T/2\) — the leads do not behave as two fully independent impedances in parallel.

Section 7

What Sets the Tower-Top Voltage

A first estimate is \(V_{\text{tower}} \sim I Z_T\), but more accurately the tower-top voltage has an inductive component \(L_T\,dI/dt\) and a footing-resistance component. The surge impedance / inductance governs the initial steep-front part; the footing resistance governs the later potential rise. So the tower-top voltage is controlled together by the tower surge impedance/inductance, the footing resistance, the ground-wire surge impedance and the stroke current front time.

Section 8

Corona on Travelling Waves

Corona begins only when the conductor surface electric field exceeds the corona inception level; below that level the conductor behaves close to its normal geometrical capacitance. Once the surface gradient exceeds inception, streamers form around the conductor and effectively enlarge the conductor radius from its physical value \(r\) to a corona radius \(R_c\) — and that changes the travelling-wave behaviour. The conductor-plus-corona-sheath acts like a larger conductor on the rising front of the wave.

Section 9

Corona as Added Capacitance

Since \(C \propto 1/\ln(2h/r)\), a larger effective radius means \(\ln(2h/R_c)\) is smaller, so capacitance increases:

\[ C_{\text{corona}} = C_n + \Delta C \]

\(C_{\text{corona}}\): effective capacitance during corona on the rising front
\(C_n\): normal (geometrical) conductor capacitance before corona
\(\Delta C\): additional voltage-dependent capacitance from the corona sheath

Section 10

Effect on Velocity and Surge Impedance

With \(L\) essentially unchanged, the extra capacitance lowers both the velocity and the surge impedance:

\[ v = \frac{1}{\sqrt{LC}}\ \Rightarrow\ C\uparrow \Rightarrow v\downarrow \qquad Z = \sqrt{\frac{L}{C}}\ \Rightarrow\ C\uparrow \Rightarrow Z\downarrow \quad (Z_c < Z_0) \]

Because corona increases the effective capacitance while the inductance is not increased in the same proportion, the effective surge impedance on the rising front is reduced (\(Z_c < Z_0\)). And since \(v = 1/\sqrt{LC}\), the added capacitance reduces the front velocity and delays the wavefront. So corona slows the high-voltage part of the wave and reduces the conductor surge impedance — which in turn changes coupling and reflection behaviour.

Section 11

What Corona Does to the Waveform

Wavefront pushback means the rising front of the surge is delayed and stretched in time, because the corona sheath absorbs charge and raises the effective capacitance. Since the higher-voltage parts travel more slowly, the front is stretched. Corona therefore causes front distortion, reduced steepness, delay of the high-voltage parts, possible crest reduction, lower surge impedance and increased coupling — though it usually affects front steepness and arrival time more than the final crest (crest reduction depends on the surge shape, conductor geometry, corona onset level and the timing of reflections). The critical modelling point:

Corona affects the front, not the tail

Corona acts mainly while the voltage is rising. After the voltage decreases and the corona sheath collapses, the conductor response tends back toward its normal capacitance and surge impedance — so using a constant reduced surge impedance for the entire waveform can be misleading, because corona is strongly voltage-dependent and mainly a rising-front effect.

Section 12

The Tidd Test Line and Wavefront Pushback

The classic Tidd test line (an experimental full-scale test line, 7170 ft / 2185 m) measured corona on travelling surges at different distances. The findings: below inception, front distortion was small; above inception, the front was pushed back; the pushback grew with distance; apparent inception voltage fell with distance; and inception is statistical, depending on wave steepness. The effect is captured by a delay per unit distance:

\[ \Delta T = \left(\frac{\Delta T}{d}\right) d \]

Each instantaneous voltage level on the front has its own \(\Delta T/d\) — the higher the voltage, the larger the delay, which reshapes the wave.

Section 13

Using the Pushback Curves

The construction: take a point on the original front, read \(\Delta T/d\) for that voltage, multiply by the travel distance \(d\), shift the point back in time by \(\Delta T\), and repeat for other voltage points. The rebuilt front has lower steepness, a delayed crest, and sometimes a lower crest.

Crest reduction: if the surge has a short tail, the delayed front can meet the falling tail before reaching the original crest — so the peak is reduced. With a long flat tail, crest reduction is smaller.

Section 14

Charge-Voltage Curve, Models and Inception

Laboratory tests measured the conductor charge–voltage \((q\text{–}e)\) relationship, whose slope is the capacitance:

\[ C = \frac{dq}{de} = C_n + \Delta C, \qquad \Delta C = f(e,\,V_i) \]

\(e\): instantaneous conductor voltage
\(V_i\): corona inception voltage

Below inception the slope is \(C_n\); above it the slope rises as corona adds capacitance. The \(q\text{–}e\) curve depends mainly on instantaneous voltage, not on front duration — supporting a voltage-dependent capacitance model. The following are empirical or semi-empirical ways to represent corona-induced front distortion in lightning-surge calculations: several capacitance formulations exist (the Weck and CIGRE corona representations — CIGRE being the International Council on Large Electric Systems — which are essentially similar), and the inception voltage scales conceptually as \(V_i \sim E_0\,r\,Z_0/60\), with the critical gradient \(E_0\) from the CIGRE or Skilling–Dykes empirical corona expression (the latter agreeing somewhat better with the test data). For bundle conductors, the effective electrical radius is larger than a single subconductor — calculated from the bundle geometry (subconductor radius, number of subconductors and spacing), not simply \(n\,r\) — and the surface electric field is generally lower, so corona onset and capacitance differ from a single conductor, usually delaying inception (one reason EHV/UHV lines use bundles).

Section 15

Corona and the Coupling Factor

To avoid clashing with the capacitances \(C_n\) and \(\Delta C\), the coupling factor is written \(k\) here (the same quantity called \(C\) on the coupling page). It uses the self-surge impedance: if \(Z_m\) is assumed approximately unchanged by corona, the reduction in self-surge impedance \(Z_0 \to Z_c\) increases the simplified coupling factor:

\[ k_0 = \frac{Z_m}{Z_0}, \quad k_c = \frac{Z_m}{Z_c}, \qquad Z_c < Z_0 \Rightarrow k_c > k_0 \]

\(k_0\) / \(k_c\): coupling factor before / during corona
\(Z_m\): mutual surge impedance (assumed roughly unchanged by corona)

During a tower or shield-wire stroke, stronger coupling raises the phase-conductor voltage in the same polarity as the ground-wire/tower voltage (\(V_{\text{phase}} = k\,V_{gw}\)), so the insulator stress is reduced approximately as \(V_{\text{ins}} \propto (1-k)\). The benefit is gentler than the raw coupling increase: e.g. \(k\) rising \(0.30 \to 0.48\) changes \((1-k)\) only \(0.70 \to 0.52\). If the original coupling is already significant, a further increase gives only a limited extra reduction — corona modifies the stress, it does not eliminate backflashover risk.

Table 1 — Corona effect at high voltage (illustrative, ~2000 kV).

Quantity	No Corona	With Corona
Surge impedance	~477 Ω	~299–348 Ω (model-dependent)
Coupling-factor ratio \(k_c/k_0\)	1.00	~1.37–1.60
Stress ratio \((1-k_0)/(1-k_c)\), \(k_0 = 0.30\)	1.00	~1.19–1.34

Modelling warning

In simplified travelling-wave correction methods, corona-reduced \(Z_c\) and corona-increased \(k_c\) are applied mainly on the rising front, where the voltage exceeds inception. In detailed EMTP® corona models the recovery toward \(Z_0\) and \(k_0\) is not instantaneous — it depends on the chosen model and its charge–voltage dynamics — so a single constant reduced surge impedance over the whole waveform can be misleading.

Section 16

Memory Map and Summary

Table 2 — Symbol reference.

Symbol	Meaning
\(Z_T\)	Equivalent tower surge impedance
\(Z_e\) / \(Z_m\)	Equivalent impedance of two tied downleads / mutual surge impedance
\(h\) / \(r\) / \(c\)	Effective tower height / equivalent radius / speed of light
\(Z_0\) / \(Z_c\)	Surge impedance before / during corona
\(C_n\) / \(\Delta C\)	Normal conductor capacitance / added corona capacitance
\(C_{\text{corona}}\)	Effective capacitance during corona
\(k_0\) / \(k_c\)	Coupling factor before / during corona
\(V_{\text{ins}}\)	Voltage across the insulator string
\(V_{\text{tower/crossarm}}\) / \(V_{\text{phase}}\)	Tower-side / induced phase-conductor voltage

Equation Summary

Tower surge impedance

\(\displaystyle Z_T \approx 60\,\ln\frac{2h}{r}\)

Two-downlead equivalent

\(\displaystyle Z_e = \frac{Z_T + Z_m}{2}\)

Surge impedance (no corona)

\(\displaystyle Z_0 = \sqrt{\frac{L}{C_n}}\)

Surge impedance (corona)

\(\displaystyle Z_c = \sqrt{\frac{L}{C_n + \Delta C}}\)

Coupling before corona

\(\displaystyle k_0 = \frac{Z_m}{Z_0}\)

Coupling during corona

\(\displaystyle k_c = \frac{Z_m}{Z_c}\)

Insulator stress

\(\displaystyle V_{\mathrm{ins}} \propto (1 - k)\)

Key messages

Tower surge impedance is a practical engineering equivalent from field behaviour, not a fixed resistance — cylinder/cone formulas suffice because backflashover is usually less sensitive to \(Z_T\) than to footing resistance, current steepness, span reflections and coupling (though \(Z_T\) still needs a reasonable value).
The meaningful quantity is the voltage between crossarm and phase conductor, where the insulator is stressed (\(V_{\text{ins}} = V_{\text{tower/crossarm}} - V_{\text{phase}}\)).
Corona (above its inception level) enlarges the effective radius on the rising front, adding capacitance: \(C_{\text{corona}} = C_n + \Delta C\).
More capacitance lowers velocity and surge impedance — pushing back the front and reducing steepness more than the crest.
Lower \(Z_c\) raises the coupling factor (\(k_c > k_0\)), reducing insulator stress \((1-k)\) and backflashover — but the gain is moderated by the \((1-k)\) dependence; it modifies, not eliminates, the risk.
In simplified methods, apply corona effects mainly on the rising front; the recovery toward \(Z_0\) and \(k_0\) on the tail is not instantaneous and depends on the corona model. EMTP® (Electromagnetic Transients Program) uses voltage-dependent (nonlinear) capacitance or empirical front corrections.

Screening formulas vs detailed EMT

The formulas here are mainly for interpretation and screening. Detailed studies should use EMT models with frequency-dependent line representation, suitable tower and footing models, and voltage-dependent corona where required, chosen to suit the objective (shielding performance, backflashover rate, substation surge transfer, insulation coordination). Avoid double-counting corona: if a line model or empirical method already includes corona front-distortion, adding a further independent corona correction can over-reduce the wavefront.

References

The standards, technical brochures, key papers and reference works behind this page.

J. A. Martinez-Velasco, Ed., Power System Transients: Parameter Determination. Boca Raton, FL, USA: CRC Press, 2010.
M. A. Sargent and M. Darveniza, “Tower surge impedance,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-88, no. 5, pp. 680–687, May 1969.
M. Ishii, T. Kawamura, T. Kouno, E. Ohsaki, K. Shiokawa, K. Murotani, and T. Higuchi, “Multistory transmission tower model for lightning surge analysis,” IEEE Transactions on Power Delivery, vol. 6, no. 3, pp. 1327–1335, Jul. 1991.
IEEE Std 1243-1997, IEEE Guide for Improving the Lightning Performance of Transmission Lines. New York, NY, USA: IEEE, 1997.
CIGRE Working Group 33.01, Guide to Procedures for Estimating the Lightning Performance of Transmission Lines, Technical Brochure 63. Paris, France: CIGRE, 1991.
CIGRE Working Group 33.07, Guidelines for the Evaluation of the Dielectric Strength of External Insulation, Technical Brochure 72. Paris, France: CIGRE, 1992.
A. R. Hileman, Insulation Coordination for Power Systems. New York, NY, USA: Marcel Dekker, 1999.

Part 8 Reading now

Travelling Waves, Tower Surge Impedance & Corona

Travelling-wave behaviour on lines and towers, surge impedance and the corona effect on the wavefront.

Series progress 8 of 16

Part 9

Lightning & Switching Current Waveforms

Choosing the stroke or surge waveform — front time, maximum steepness and the analytical representations.