EMTP® Overhead Lines

Solution of Overhead Line Equations in EMTP®

Once the geometry and material data are in, EMTP® has to turn the per-unit-length \(Z'(\omega)\) and \(Y'(\omega)\) matrices into a time-domain model that launches, delays, reflects, attenuates and couples travelling waves. This guide — the companion to overhead-line modelling — explains what the line model does internally: the propagation constant and characteristic admittance, modal versus phase-domain solutions, constant-parameter and frequency-dependent fitting, and how to read and sanity-check the Line Constants output before you trust it.

Reading time ≈ 23 min · EMTP® overhead-line guide

After the overhead-line geometry and material data are defined, the next step is to convert the line parameters into a time-domain model EMTP® can use. This matters because an overhead line is a distributed electromagnetic system: voltage and current do not appear instantly at the far end — they travel as waves, reflect at discontinuities, attenuate with distance and couple between phases. You do not need to derive the solution by hand, but understanding what the model does internally helps you choose the right model, read the Line Constants output, and avoid using a model outside its valid range.

This page concentrates on the solution of the line equations and on the Line Constants routine. The physical modelling decisions — lumped versus distributed, what to include for each transient, earth return, skin effect, shield wires, towers and corona — are covered on the overhead-line modelling page; here we cross-link them rather than repeat them.

Abbreviations used on this page

EMTP®Electromagnetic Transients Program

\(Z'\) / \(Y'\)Per-unit-length series impedance / shunt admittance

\(\Gamma\)Propagation constant matrix

\(Y_c\)Characteristic admittance matrix

\(H\)Propagation function, \(e^{-\Gamma\ell}\)

\(\ell\)Line length

CP / FDConstant-parameter / frequency-dependent line model

ULMUniversal Line Model (FD phase-domain)

LCCLine Constants routine

EMTElectromagnetic transient

ARMAAutoregressive moving-average (fitting)

FFTFast Fourier Transform

Key idea

The solution turns \(Z'(\omega)\), \(Y'(\omega)\) into a travelling-wave model described by the propagation constant \(\Gamma\) and characteristic admittance \(Y_c\) — it stores wave history and injects history-dependent currents at each line end.
Modal-domain solutions are efficient and fine for symmetrical / transposed lines; for untransposed, multicircuit or strongly coupled lines a phase-domain solution is usually more reliable.
Positive-sequence quantities are relatively stable with frequency; zero-sequence and ground-mode quantities are far more sensitive to frequency, earth return and soil resistivity.
Read the Line Constants output before simulating — sequence impedances, surge impedance, velocity and attenuation should all be physically reasonable. Strange output usually means a geometry, phase or data error.

Key terms used on this page

01Propagation constant, \(\Gamma\)

\(\Gamma=\sqrt{Z'Y'}\); its real part is attenuation and its imaginary part is phase shift, per unit length, for each mode.

02Characteristic admittance, \(Y_c\)

\(Y_c=Z'^{-1}\Gamma\); the travelling-wave ratio of current to voltage seen at a line end.

03Propagation function, \(H\)

\(H=e^{-\Gamma\ell}\); how a wave is delayed and attenuated over the line length \(\ell\).

04History term / convolution

The time-domain memory that carries past travelling-wave history into the present terminal currents.

05Aerial mode

A propagation mode largely confined to the conductors — fast (near \(c\)) and little affected by earth return.

06Ground mode

A propagation mode that returns through the earth — slower and strongly frequency- and soil-dependent.

07Modal transformation

The matrix that decouples phase quantities into modes; constant for ideal / transposed lines, frequency-dependent otherwise.

08Rational fitting

Approximating \(Y_c\) and \(H\) by stable rational functions so they can be applied by recursive convolution in time.

09Passivity

The property that a line model cannot generate energy; required for a numerically stable simulation.

10Line Constants routine

The EMTP® tool that converts geometry and material data into \(Z'\), \(Y'\) and the chosen line model.

11Positive / zero sequence

Symmetrical-component impedances; the zero-sequence (ground) mode is dominated by earth return.

12Nonuniform line

A line whose geometry or parameters change along its length — crossings, entrances and special spans.

Section 1

From parameters to a travelling-wave model

In the frequency domain an overhead line is described by two per-unit-length matrices — the series impedance \(Z'(\omega)\) and the shunt admittance \(Y'(\omega)\):

\[ Z'(\omega)=R'(\omega)+j\omega L'(\omega) \qquad Y'(\omega)=G'+j\omega C' \]

\(Z'\) / \(Y'\): series impedance / shunt admittance per unit length
\(R',\ L',\ G',\ C'\): resistance, inductance, conductance, capacitance per unit length
\(\omega\): angular frequency, rad/s

These are produced by the Line Constants routine from the geometry and material data (see overhead-line modelling). The solution turns them into a propagation model.

From these, the software computes how waves travel. In the time domain the frequency-domain relationships appear as convolutions — in plain terms, the present response depends on the stored past response of the line, not only on the instantaneous voltage: the present terminal current depends not only on the present voltage but on the past travelling-wave history at both ends. That is why a distributed line model carries history terms (history current sources), and why reflections and propagation delay are included naturally. Because the line travel time is rarely a whole number of time-steps, the model interpolates the stored history between steps — a detail that matters for accuracy at small time-steps. Frequency-dependent models need more numerical work because the parameters vary with frequency — the main reason an overhead-line model is more than a PI circuit.

Section 2

Characteristic admittance and the propagation function

Two functions describe almost everything a time-domain line model does — a local one and a travelling-wave one.

\[ \Gamma(\omega)=\sqrt{Z'(\omega)\,Y'(\omega)} \qquad Y_c(\omega)=Z'(\omega)^{-1}\,\Gamma(\omega) \]

\(\Gamma\): propagation constant matrix — attenuation (real part) and phase shift (imaginary part) per unit length
\(Y_c\): characteristic admittance matrix — the travelling-wave current-to-voltage ratio at a line end

Read these as compact engineering notation, not ordinary scalar algebra. For a scalar line they reduce to the familiar \(\Gamma=\sqrt{Z'Y'}\) and \(Y_c=\sqrt{Y'/Z'}=1/Z_c\). For a multiconductor line, EMTP® obtains the modal propagation constants and the characteristic admittance through eigen/modal decomposition or phase-domain fitting; because the matrices do not commute, the compact forms above stand for those procedures rather than literal scalar algebra. \(Y_c\) is a characteristic admittance, not an impedance: the characteristic impedance is \(Z_c=Y_c^{-1}\) (\(Z_c=1/Y_c\) only in the scalar case).

\[ H(\omega)=e^{-\Gamma(\omega)\,\ell} \]

\(H\): propagation function — delay and attenuation across the line
\(\ell\): line length, m

\(H\) is the heart of a distributed model: the line is not just an impedance, it is a propagation path, so the longer the line the more delay, attenuation and phase shift the wave sees. \(Y_c\) is the local terminal voltage–current relationship — smoother and generally easier to fit than \(H\), the travelling-wave part, which is harder because different modes travel at different speeds and attenuate differently. Before fitting, the bulk modal time delay is usually factored out of \(H\) so the remainder is minimum-phase and easier to approximate. Keep \(\ell\) consistent with the per-unit-length basis of \(Z'\) and \(Y'\): if those are quoted per kilometre then \(\ell\) is in km, if per metre then \(\ell\) is in metres.

For a multiphase line the propagation is not one wave at one velocity but several modes — combinations of phase and ground-mode behaviour. Aerial modes travel fast, near the speed of light, and are little affected by earth return; ground modes are slower and more attenuated in the power-transient band, and strongly affected by soil resistivity and frequency-dependent earth return. This is why the propagation function is usually the difficult part of a frequency-dependent overhead-line model.

Section 4

Phase-domain solution

Phase-domain techniques solve the line equations directly in phase quantities, avoiding the frequency-dependent modal-transformation problem. Working directly with phase voltages and currents is more robust for untransposed, asymmetrical and multicircuit lines and where the modal transformation is strongly frequency-dependent — it represents frequency-dependent coupling between physical conductors directly. The cost is a more demanding numerical implementation: the propagation functions are harder to fit, and the model needs careful passivity, stability and frequency-range checks.

How phase-domain models are built

Modern phase-domain models — the Universal Line Model (Morched, Gustavsen and Tartibi) and Noda’s ARMA approach among them — use rational fitting, recursive convolution and s- or z-domain techniques to turn the frequency-dependent line behaviour into a stable, efficient time-domain model. You do not need to memorise the numerical methods, but you should know their purpose and check the result for passivity, causality and stability.

Which to use

For simple, symmetrical lines a modal-domain model may be enough. For complex, untransposed or strongly coupled lines a phase-domain model is usually more reliable. For wideband EMT studies, check the chosen model for stability, passivity and validity over the relevant frequency range.

Section 5

Frequency-dependent and constant-parameter solutions

A frequency-dependent model represents how resistance, inductance and earth-return effects vary with frequency — an overhead line does not have one impedance from 50 Hz to lightning frequencies. The effect is most visible in ground-mode and zero-sequence behaviour: positive-sequence inductance is often stable over a wide range, but zero-sequence resistance and inductance vary strongly because they depend on earth return. For switching and lightning, a frequency-dependent model can give a noticeably different wave shape — more attenuation and front spreading — than a constant-parameter model. That is physics (skin effect and frequency-dependent earth return), not a numerical artefact.

A constant-parameter distributed model uses parameters at one selected frequency. It still represents the line as a travelling-wave system, but not the variation with frequency. It can be adequate when the study sits in a narrow band — for example a fast-front study using a representative high frequency over a short section — but one frequency cannot represent a broadband surge, and an unrepresentative choice gives misleading attenuation or velocity. A constant-parameter model also holds the modal transformation matrix constant — a second reason it is inadequate for broadband or asymmetrical lines, and part of why frequency-dependent (J. Marti) and phase-domain (ULM) models exist.

Choosing the solution

Use constant-parameter distributed models when the band is narrow or a representative frequency is justified; use frequency-dependent models when damping, ground-mode propagation or broadband accuracy matters; and do not use a 50/60 Hz parameter set for a fast transient unless the simplification is clearly justified. The conceptual basis is on the modelling page.

Section 6

Line Constants routine: input data

Most EMTP® programs include a Line Constants routine that converts physical line data into electrical parameters and then into the chosen model. The inputs are mostly geometrical and must be entered carefully — conductor and shield-wire coordinates, phase and circuit designation, bundle spacing and orientation, sag or average height, transposition, conductor diameter or radius, DC resistance or resistivity, shield-wire resistance, ground resistivity, and the model type. Errors in tower drawings, phase coordinates, bundle arrangement or shield-wire position feed straight into the calculated matrices.

Two inputs that need judgement

Conductor resistance is not a fixed catalogue number — it depends on temperature, so model at operating temperature where it matters. Ground resistivity is usually the least certain input; it varies along the route and with soil condition. For low-frequency, balanced studies a moderate error may not dominate, but for high-frequency, zero-sequence, ground-mode and induced-voltage studies it needs care (Section 9). Conductor heights also follow the software’s convention — some routines expect the tower attachment height and sag, others the average conductor height — so enter what the routine asks for.

Section 7

Line Constants routine: output and sanity checks

The routine’s outputs are a check on the model before a full simulation, and should not be ignored: the series-impedance and shunt-capacitance/susceptance matrices, positive- and zero-sequence resistance and inductance, surge impedance, attenuation, propagation velocity and wavelength — at a frequency or over a range.

Do the positive- and zero-sequence impedances look reasonable?
Is the zero-sequence resistance much more frequency-dependent than the positive-sequence resistance?
Is the calculated capacitance consistent with the line geometry?
Is the surge impedance physically reasonable — the aerial / phase mode is typically in the order of a few hundred ohms (lower for bundled EHV, higher for a single thin conductor), with the earth-return mode higher?
Is the propagation velocity physically reasonable — at or below the speed of light (aerial modes near \(c\), the ground mode slower)?
Are the \(Z'\) and \(Y'\) matrices symmetric (reciprocal), with per-unit-length \(R\), \(L\) and \(C\) all positive?
Does the frequency response change smoothly, with no unexpected numerical behaviour?

If the output looks physically strange, the error is usually in the conductor coordinates, phase assignment, bundle data, conductor resistance, ground resistivity or transposition setup — not in the simulation.

Section 8

Where accuracy matters: parameter sensitivity

Overhead-line parameters do not respond equally to every input uncertainty, and knowing where accuracy matters most saves effort and avoids false confidence.

Capacitance is geometry-driven and essentially frequency-independent over the design range — get the geometry right, but it does not vary with frequency.
Positive-sequence inductance is relatively stable with frequency; zero-sequence resistance and inductance are much more sensitive, because they depend on earth return.
Conductor resistance matters more at high frequency through skin effect.
Ground resistivity matters far more for zero-sequence and ground-mode behaviour than for balanced positive-sequence behaviour.
Conductor height and spacing must be entered correctly, but moderate variations within normal transmission geometry may not dominate the result.
A frequency-dependent model can produce significant attenuation and front distortion even over moderate line lengths — which is why CP and FD models can give visibly different switching or step responses.

Spend effort where the sensitivity is

This is not a licence to be careless with geometry. It means that, in many practical cases, the result is more sensitive to frequency dependence, earth return and sequence mode than to small geometry variations — so concentrate the effort there.

Section 9

Ground resistivity in practice

Ground resistivity is among the most uncertain inputs — it varies along the route and with depth, moisture, season and soil type — so the practical question is how much that uncertainty affects the study. For balanced positive-sequence studies the effect is often limited. For zero-sequence, ground faults, ground-mode transients, shield-wire effects and induced-voltage studies it is much more important. For high-frequency transients it can influence attenuation, propagation velocity and the dominant oscillation frequency. Higher resistivity deepens the earth-return current path, which raises the ground-mode series inductance and so slows the ground mode and increases its attenuation, shifting the dominant frequency and altering receiving-end and induced overvoltages. The aerial (positive-sequence) mode travels near the speed of light and is essentially insensitive to soil resistivity — it is the ground mode that responds.

How much effort to spend

Do not over-refine ground resistivity for low-frequency balanced studies that are not sensitive to it. Do check its sensitivity for zero-sequence energisation, earth faults, induced voltages, lightning and open-ended line overvoltages. Run a sensitivity check where the route includes very high-resistivity ground, and if the result moves significantly when resistivity is varied, document the value chosen and its basis.

Section 10

Ground (shield) wires and ground-mode behaviour

Ground wires are not only lightning-shielding conductors — in the line solution they also change the ground-mode and zero-sequence propagation path. They provide lightning shielding, change the electromagnetic coupling between phases and earth, and influence zero-sequence and ground-mode behaviour. For lightning they are part of the current path and interact with tower surge impedance and footing impedance (see overhead-line modelling). For switching and low-frequency studies they may or may not matter. A useful observation: removing the ground wires makes the transient response more sensitive to ground resistivity, because the ground wire is an additional metallic return path that changes the ground-mode behaviour.

When to include them

Include ground wires when lightning performance, tower voltage, backflashover, ground-mode propagation or induced voltage matters — and do not assume they are irrelevant just because the study is not a direct strike. If the transient has a strong zero-sequence or common-mode component, check whether the ground wires change the result.

Section 11

Sequential energisation and switching

Sequential pole closing or phase-by-phase energisation produces unbalanced conditions even from a balanced source, because the phases do not close at the same instant. Here line asymmetry, frequency-dependent parameters, ground resistivity and ground wires can all affect the overvoltage — not only its peak, but its oscillation frequency, damping and phase coupling. Sequential closing can excite zero-sequence and ground-mode components, and a frequency-dependent model can give different damping and wave shape from a constant-parameter one.

Do not assume positive-sequence

For three-phase switching studies, do not assume a purely positive-sequence line response. This matters most for open-ended line energisation, transformer energisation through long lines, shunt-reactor switching and controlled (point-on-wave) switching.

Section 12

Corona in the solution

Corona affects the solution because it turns the line into a nonlinear, distributed problem: above the conductor-surface inception field it adds voltage-dependent capacitance and loss, so the line may need segmentation into short sections with nonlinear shunt branches, solved iteratively at each time-step. Frequency-dependent line modelling handles the linear propagation and earth-return effects; corona is a separate nonlinear mechanism and should not be assumed to be included unless the selected model explicitly represents it. A model without corona can be conservative for some propagated-surge studies (it may over-estimate peak and steepness), but that is not a blanket rule — it depends on the waveform, line length, voltage, weather, geometry and corona model. The detailed EMTP® corona implementation — device parameters, segmentation, the charge–voltage curve and validation — is on the dedicated corona modelling page; if the result is close to the insulation withstand limit and corona is expected, state the assumption.

Section 13

Nonuniform lines

Most line models assume the line is uniform along its length. In reality the geometry can change — near river or long-span crossings, substation entrances, gantries, cable sealing ends, compact sections, transpositions, special towers and places where conductor height changes significantly. There, the longitudinal variation of parameters affects surge propagation and reflections, and a single uniform section may not be accurate.

Two ways to handle it

Subdivide the line into several uniform sections, or use a method that handles distance-dependent parameters directly (a finite-difference representation is one option where the line must be discretised anyway). Careless subdivision can introduce spurious numerical oscillations if a section length is not matched to its travel time, so use a uniform model for ordinary spans, a segmented or nonuniform representation where geometry changes, and take particular care near substations and special crossings — common sites of reflection and insulation stress.

Section 14

Frequency-domain solution methods

Some studies solve the line equations directly in the frequency domain and transform back with a numerical inverse — the Fast Fourier Transform, or, more accurately, the numerical Laplace transform (essentially an FFT with a damping coefficient that suppresses Gibbs oscillation and aliasing). Because frequency-domain superposition assumes linearity, it is used mainly for benchmarking, comparing line-constant formulations and linear network response — not for switching with nonlinear elements such as arresters or corona. For day-to-day work the built-in time-domain model is used, but the frequency-domain view is a useful reminder that every transient has frequency content: a waveform is not defined by one frequency, the chosen model must be valid over the band that contributes to it, and a model fitted over the wrong range gives an inaccurate time-domain waveform.

Section 15

Workflow and pre-acceptance checks

A practical sequence: identify the study type (temporary overvoltage, switching, lightning, induced voltage, power quality, secondary arc, energisation, reclosing or incoming surge); estimate the frequency range (low-frequency studies need a wide network, fast-front studies need detailed local representation); select the model type (PI for low-frequency or short lines, distributed where propagation and reflection matter, frequency-dependent where damping or the ground mode matters); prepare the geometry and the conductor / ground data carefully; review the Line Constants output; decide whether shield wires, towers, footing impedances, insulators, flashover paths, corona or nonuniform sections are needed; run sensitivity checks where uncertainty matters (ground resistivity, corona, footing resistance, frequency dependence); and document the assumptions, especially for insulation coordination, arrester duty, switching-overvoltage or lightning-performance work.

Before accepting the model, confirm:

The line length and the model’s frequency range suit the transient.
The conductor coordinates, phase order, bundle and transposition data are correct.
Shield wires are included or excluded intentionally.
The conductor resistance corresponds to the intended temperature.
The ground resistivity is reasonable, with a sensitivity check where it matters.
The positive- and zero-sequence outputs, surge impedance and propagation velocity are physically reasonable.
The frequency response is smooth, with no unexpected numerical behaviour.
Corona and nonuniform sections are included where they control the result.
The terminal network is detailed enough for realistic reflections.

Section 16

Main takeaway

Solving the line equations is not about the mathematics — it produces a model whose characteristic admittance and propagation function set how waves are launched, delayed, reflected, attenuated and coupled between phases. Modal-domain models suit symmetrical lines but lose accuracy when the modal transformation is strongly frequency-dependent, while phase-domain models are more robust for complex, untransposed and strongly coupled lines at the cost of harder fitting; the Line Constants output should be reviewed rather than trusted blindly, since the zero-sequence and ground-mode quantities are the frequency-sensitive ones. The line model is acceptable only if it reproduces the propagation delay, damping, reflection and phase coupling required by the transient.

References

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

H. W. Dommel, Electromagnetic Transients Program Reference Manual: EMTP Theory Book. Portland, OR, USA: Bonneville Power Administration, 1986.
J. R. Marti, “Accurate modelling of frequency-dependent transmission lines in electromagnetic transient simulations,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-101, no. 1, pp. 147–157, Jan. 1982.
A. Semlyen and A. Dabuleanu, “Fast and accurate switching transient calculations on transmission lines with ground return using recursive convolutions,” IEEE Transactions on Power Apparatus and Systems, vol. PAS-94, no. 2, pp. 561–571, Mar./Apr. 1975.
A. Morched, B. Gustavsen, and M. Tartibi, “A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables,” IEEE Transactions on Power Delivery, vol. 14, no. 3, pp. 1032–1038, Jul. 1999.
B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Transactions on Power Delivery, vol. 14, no. 3, pp. 1052–1061, Jul. 1999.
J. A. Martinez-Velasco, Ed., Power System Transients: Parameter Determination. Boca Raton, FL, USA: CRC Press, 2010.

Part 6 Reading now

Overhead Line Equations & Solution

The line equations, modal and phase-domain solution, and frequency-dependent line fitting.

Series progress 6 of 16

Part 7

Corona Effect in EMTP® Overhead Lines

How corona attenuates and distorts travelling surges, and the q–v and distributed nonlinear models.