Chapter 11 — Frequency-dependent line models

Every line model presented so far has assumed that $R^{\prime }$, $L^{\prime }$, and $C^{\prime }$ are constants. They are not. Current crowds toward the conductor surface as frequency rises, and the earth-return path that carries the ground mode changes character completely across the band a transient occupies. A model built from one set of constants is a model tuned to one frequency, and the further a study strays from that frequency, the more the answer drifts.

This chapter explains what the constant-parameter model gets wrong and how a frequency-dependent model is built. It is the most demanding material in the module, and it yields not only an understanding of the better model but a precise account of which results the simpler one can still be trusted to deliver.

Not yet in NumaSim

NumaSim's transmission line is a constant-parameter travelling-wave model; no frequency-dependent line exists today. The chapter is included because the limitation is real and must be reasoned about, and because §11.9 converts the theory into concrete guidance on parameter choice and result interpretation that applies to the present implementation.

Learning objectives

By the end of this chapter you should be able to:

• Describe how $R^{\prime }$ and $L^{\prime }$ vary with frequency, and why the ground mode varies most.
• Write the exact frequency-domain line equations in terms of $Z_C(\omega )$ and the propagation function $A(\omega )$.
• Explain why the exact time-domain model is a convolution, and why a direct implementation is unaffordable.
• Explain how rational fitting plus recursive convolution reduce the cost to a fixed number of operations per step.
• Explain why the pure delay must be extracted before fitting, and why the fitting band cannot start at DC.
• State the trade-off between modal-domain and phase-domain fitting.
• Choose constant parameters appropriate to the phenomenon under study, and identify which results the choice compromises.

11.1 What varies, and by how much

Two mechanisms make a line's parameters frequency-dependent.

Skin effect in the conductor. At DC, current fills the conductor uniformly. As frequency rises it concentrates near the surface, reducing the effective cross-section. The internal resistance therefore rises — asymptotically as $\sqrt{f}$ — and the internal inductance falls, because less magnetic flux links the interior. For an aerial mode this is the whole story, and it is a modest effect: most of the loop inductance is external, so $L_1$ barely moves.

The earth return. For the ground mode, current returns through the soil, and the depth at which it does so shrinks as frequency rises. The consequences are large. At low frequency the earth contribution to the series resistance is

$$R_{\text{earth}}^{\prime }\approx \frac{\omega {\mu }_0}{8},$$

which is remarkable for what it does not contain: soil resistivity drops out entirely, and the resistance is directly proportional to frequency. At 60 Hz this adds about $0.06\mathrm{\Omega }/\text{km}$, comparable to the conductor's own resistance. At 5 kHz it is nearly $5\mathrm{\Omega }/\text{km}$ — roughly a hundred times larger.

That single fact is the most important practical consequence of this chapter. Zero-sequence resistance rises steeply with frequency and positive-sequence resistance barely does. A model whose $R_0$ was taken at power frequency will substantially under-damp any transient with zero-sequence content — which means single-phase faults, single-pole switching, and unbalanced energization.

The surge impedances follow along:

Sketch of how modal surge impedance varies with frequency. The aerial mode is nearly constant across the whole band; the ground mode falls steeply and grows without bound as the frequency approaches DC. Values are illustrative of a typical overhead line, not a specific one.

11.2 The line in the frequency domain

Working one frequency at a time, the coupled slice equations of Chapter 8 solve exactly. Define the propagation constant and the characteristic impedance

$$\gamma (\omega )=\sqrt{Z^{\prime }(\omega )Y^{\prime }(\omega )}=\alpha (\omega )+j\beta (\omega ),Z_C(\omega )=\sqrt{\frac{Z^{\prime }(\omega )}{Y^{\prime }(\omega )}},$$

with $Z^{\prime }=R^{\prime }+j\omega L^{\prime }$ and $Y^{\prime }=G^{\prime }+j\omega C^{\prime }$. Both are now complex and both depend on frequency. Over a line of length $d$ the propagation function is

$$A(\omega )=e^{-\gamma (\omega )d}=\underset{\text{attenuation}}{\underbrace{e^{-\alpha d}}}\cdot \underset{\text{phase shift}}{\underbrace{e^{-j\beta d}}}.$$

The travelling-wave relation of Chapter 8 survives intact, with $A$ standing in for the pure delay. Keeping the same sign convention — currents into the line at each end — the wave launched at $k$ arrives at $m$ multiplied by $A$:

$$V_m(\omega )-Z_C(\omega )I_{mk}(\omega )=A(\omega )[V_k(\omega )+Z_C(\omega )I_{km}(\omega )].$$

Setting $R^{\prime }=G^{\prime }=0$ recovers Chapter 8 exactly: $Z_C$ becomes the real constant $\sqrt{L^{\prime }/C^{\prime }}$, $\alpha$ vanishes, $\beta =\omega \sqrt{L^{\prime }{C}^{\prime }}$, and $A=e^{-j\omega \tau }$, a phase shift proportional to frequency, which is the frequency-domain signature of a pure time delay.

Everything that makes a real line different from Chapter 8's is contained in the two ways $A$ departs from that ideal:

• $|A|=e^{-\alpha d}<1$, and $\alpha$ grows with frequency. High frequencies are attenuated more than low ones, so a step front does not merely shrink — it rounds off.
• $\beta /\omega$ is not constant, so different frequencies travel at different speeds. This is dispersion, and it smears a sharp disturbance out in time.

A step launched down a real line. The constant-parameter model delivers a perfect copy delayed by $\tau$. A frequency-dependent line attenuates the high-frequency content more than the low, so the front arrives rounded, reduced, and slightly late.

11.3 From multiplication to convolution

Rearranging the frequency-domain relation into the Norton form we have used since Chapter 2, with $Y_C=1/Z_C$:

$$I_{mk}(\omega )=Y_C(\omega )V_m(\omega )-A(\omega )[Y_C(\omega )V_k(\omega )+I_{km}(\omega )].$$

Structurally this is Chapter 8: a term in the local voltage plus a term arriving from the far end. Every product of frequency-domain quantities is, however, a convolution in the time domain:

$$i_{mk}(t)=y_c(t)\ast v_m(t)-a(t)\ast [y_c(t)\ast v_k(t)+i_{km}(t)],$$

where $y_c(t)$ and $a(t)$ are the impulse responses of $Y_C(\omega )$ and $A(\omega )$. Each convolution is an integral over the entire past of its input.

A literal implementation is not viable. The response $a(t)$ can last many travel times, so at $\mathrm{\Delta }t=20\mu \text{s}$ the sum may run to thousands of products per line end per step, and the cost per step grows with the reach of the response. No practical model is built this way. What makes frequency-dependent lines tractable is a pair of ideas that together reduce the cost to a small, fixed number of operations per step.

11.4 Rational approximation

The first idea is to stop treating $A(\omega )$ and $Z_C(\omega )$ as arbitrary functions and approximate each by a ratio of polynomials — in practice, a sum of simple first-order terms:

$$A(\omega )\approx \sum _{i=1}^n\frac{c_i}{j\omega +p_i}.$$

The fit is done numerically against values computed from the physical line data, over the frequency band the study cares about. Two constraints matter: all poles $p_i$ must have positive real parts, or the model is unstable; and the fitted function must remain passive, or the model can manufacture energy.

This particular form is chosen for what it becomes in the time domain. Each term $c_i/(j\omega +p_i)$ transforms to a decaying exponential $c_i{e}^{-p_{i}t}$, so the impulse response becomes a sum of exponentials, and convolution with an exponential can be performed recursively.

11.5 Recursive convolution

Consider one exponential term:

$$y(t)={\int }_0^{t}c{e}^{-pu}x(t-u)du.$$

Split the integral at $u=\mathrm{\Delta }t$. The part beyond $\mathrm{\Delta }t$ is, by inspection, $e^{-p\mathrm{\Delta }t}$ times the same integral evaluated one step earlier. With $x$ assumed to vary linearly across the step, the whole convolution therefore collapses to a two-term recursion:

$$y(t)=my(t-\mathrm{\Delta }t)+c_{0}x(t)+c_{1}x(t-\mathrm{\Delta }t),m=e^{-p\mathrm{\Delta }t},$$

with $c_0$ and $c_1$ fixed constants derived from $c$, $p$, and $\mathrm{\Delta }t$. One multiply-accumulate per term, per step. The entire infinite history is carried in a single state variable $y(t-\mathrm{\Delta }t)$.

This recursion is the load-bearing step of the whole subject. An $n$-pole fit costs $n$ recursions per step, typically fewer than twenty, irrespective of the line's length, its travel time, or the duration of its impulse response. The convolution ceases to be an integral over the past and becomes a handful of scalars updated in place — the same manoeuvre that turned the inductor's differential equation into a companion model in Chapter 2.

11.6 Extracting the delay

One complication remains, and neglecting it is the classic route to a fit that will not converge.

The propagation function contains a pure delay. Even on a heavily lossy line, nothing arrives before ${\tau }_{min}=d\sqrt{L_{\mathrm{\infty }}^{\prime }{C}^{\prime }}$, the travel time computed with the high-frequency inductance. In the frequency domain this appears as a factor $e^{-j\omega {\tau }_{min}}$, a phase that decreases without bound as frequency rises. A rational function of finite order cannot reproduce unbounded phase; a fitting routine asked to attempt it will consume every available pole and still fail.

The fix is to take the delay out first:

$$A(\omega )=e^{-j\omega {\tau }_{min}}{A}^{\prime }(\omega ).$$

The remaining $A^{\prime }(\omega )$ is smooth, well-behaved, and fits with a modest number of poles. The delay is then reinstated with exactly the ring buffer of Chapter 8. Viewed in the time domain, the operation is a shift: the impulse response is wound back so it begins at $t=0$ instead of $t={\tau }_{min}$, fitted there, then wound forward again at run time.

Structure of a frequency-dependent line end. The pure delay is handled by a buffer exactly as in Chapter 8; only the smooth remainder is fitted, and its convolutions are evaluated by first-order recursions. The result is a Norton equivalent of the same shape as every companion model since Chapter 2.

11.7 The low-frequency trap

A second practical difficulty lives at the other end of the band. Since

$$Z_C(\omega )=\sqrt{\frac{R^{\prime }+j\omega L^{\prime }}{G^{\prime }+j\omega C^{\prime }}}$$

and the shunt conductance $G^{\prime }$ of an overhead line is essentially zero, the denominator vanishes as $\omega \to 0$ and $Z_C$ grows without bound, roughly as $1/\sqrt{\omega }$. That is the rising left-hand tail in the first figure of this chapter, and it is genuine physics — at DC there is no shunt path, so the line cannot present a finite surge impedance.

It is also ruinous for a rational fit. A fitting routine asked to cover DC spends its poles chasing an asymptote and returns a mediocre fit across the band of interest.

Two remedies are standard, and both are approximations that should be recognized as such:

• Start the fitting band above DC, at a fraction of a hertz or a few hertz. Everything below is extrapolated.
• Add a small artificial shunt conductance so that $Z_C$ remains finite all the way down.

Under either remedy the model's true-DC behaviour is not guaranteed by the fit, so any study whose answer depends on it — trapped charge on a de-energized line, or slow DC decay after a switching operation — warrants an independent check rather than trust.

11.8 Modal or phase domain?

Chapter 10 decoupled a three-phase line by transforming to modal coordinates with a real, constant matrix. That was legitimate because the line was balanced and its parameters were constants. Neither holds here.

Modal-domain fitting applies the transformation first and then fits one scalar $A(\omega )$ and one $Z_C(\omega )$ per mode — three fits for a three-phase line, which is cheap. The difficulty is the transformation itself: for an untransposed line, and for essentially any cable, the eigenvectors are complex and vary with frequency. Forcing a single real constant matrix introduces error that no amount of care in the modal fits can recover. Worse, as frequency sweeps, two eigenvalues can approach and exchange roles, so a naive eigen-solver returns modes that swap identity mid-band and produce fitted functions with discontinuities in them. The eigenvectors have to be tracked deliberately.

Phase-domain fitting skips the transformation and fits the full $n\times n$ propagation matrix directly. That is $n^2$ functions instead of $n$, and enforcing stability and passivity on a matrix fit is harder than on a scalar one. In exchange, the entire class of transformation-matrix error disappears. For cables — where the modal matrix is strongly frequency-dependent — this is not really a choice, and phase-domain models have become the standard for them.

The general principle extends well beyond line models: a transformation that is exact for constant parameters becomes an approximation the moment those parameters vary, and whether the approximation still earns its place is a question worth asking each time.

11.9 Working within the constant-parameter model

The remainder of the chapter is directly applicable to the present implementation.

Choose parameters for the phenomenon, not for the line. No single correct $R$, $L$, $C$ exists for a line; there is only a set that is correct near a given frequency. The frequency to select is the one that dominates the transient under study:

Study	Representative frequency
Load flow, steady state, temporary overvoltage	50–60 Hz
Line energization, switching surge, fault transient	0.5–5 kHz
Restrike, breaker transient recovery voltage	5–50 kHz
Lightning, very fast front	100 kHz – 1 MHz

Line-constants data is usually reported at power frequency. Using it for a switching study means using an $R_0$ that is roughly a hundred times too small, as §11.1 showed.

Adjust the zero-sequence resistance deliberately. Where a study has zero-sequence content and the decay rate of the transient matters, enter an $R_0$ representative of the study band rather than the datasheet value. Chapter 10's Bergeron — Sequence RLC entry sets $R_0$ independently of $R_1$; the balanced-matrix entry cannot. Recomputing $R_0$ at a few kilohertz and entering that value is a crude correction, but it is far preferable to accepting a hundredfold error.

Know which results survive. A constant-parameter travelling-wave line remains correct in the following respects:

• Arrival times and the entire reflection pattern, since these depend on $\tau$ and $Z_C$ rather than on their variation.
• Surge impedance and attenuation at, and near, the chosen frequency.
• The steady-state DC voltage drop, which is exactly $R$ by construction (Chapter 9).

The following results should be treated with suspicion:

• Front shape. Real lines round a steep front substantially over a few tens of kilometres, whereas a constant-parameter model preserves it almost perfectly. Any result turning on rise time, insulation coordination in particular, therefore errs on the optimistic side.
• Damping of zero-sequence transients, for the reason given above. Ground-mode oscillations persist longer in the simulation than in the field.
• Peak overvoltage after several reflections. Errors in attenuation compound with each transit, making the tenth reflection far less trustworthy than the first.

Know when to stop. Where the answer depends on the shape of a microsecond-scale front after tens of kilometres of travel, a constant-parameter model is not the right tool at any parameter setting. Establishing that is itself a legitimate outcome of a study.

11.10 Summary

• Conductor skin effect and, far more importantly, the earth return make line parameters frequency-dependent. Zero-sequence resistance rises roughly in proportion to frequency; positive-sequence resistance barely moves.
• Exactly, a line is described by $Z_C(\omega )$ and the propagation function $A(\omega )=e^{-\gamma (\omega )d}$, whose magnitude attenuates and whose phase disperses. Setting the losses to zero recovers Chapter 8's pure delay.
• In the time domain those products become convolutions over the entire past — unaffordable if implemented literally.
• Fitting $A$ and $Z_C$ with sums of first-order rational terms turns each convolution into a two-term recursion, giving a fixed cost per step regardless of the line's travel time.
• The pure delay must be factored out before fitting, because finite-order rational functions cannot represent unbounded phase; it is restored with a buffer.
• $Z_C$ diverges as frequency approaches DC, so the fitting band starts above zero, or a small artificial shunt conductance is introduced.
• Modal fitting is cheap but inherits the error of a constant real transformation matrix; phase-domain fitting avoids that at the cost of fitting $n^2$ functions, and is effectively mandatory for cables.
• Until NumaSim gains such a model, pick constant parameters at a frequency representative of the study, correct $R_0$ upward for transient work, and treat front shape and long-run damping as the results most likely to be wrong.

11.11 Problems

Problem 11.1. Estimate the earth-return resistance contribution at 60 Hz, 1 kHz, and 20 kHz. What does the trend imply for a zero-sequence transient at a few kilohertz simulated with 60 Hz data?

Solution 11.1

Using $R_{\text{earth}}^{\prime }\approx \omega {\mu }_0/8$ with ${\mu }_0=4\pi \times {10}^{-7}\text{H/m}$:

At 60 Hz, $\omega =377$: $R^{\prime }=377(1.257\times {10}^{-6})/8=5.9\times {10}^{-5}\mathrm{\Omega }/\text{m}=0.059\mathrm{\Omega }/\text{km}$.

At 1 kHz, $\omega =6283$: $R^{\prime }=9.9\times {10}^{-4}\mathrm{\Omega }/\text{m}=0.99\mathrm{\Omega }/\text{km}$.

At 20 kHz, $\omega =1.257\times {10}^5$: $R^{\prime }=1.97\times {10}^{-2}\mathrm{\Omega }/\text{m}=19.7\mathrm{\Omega }/\text{km}$.

The relationship is linear in frequency, so a few kilohertz means one to two orders of magnitude more zero-sequence resistance than at power frequency. A simulation using 60 Hz data will show a ground-mode oscillation decaying far too slowly — the transient will ring on for many more cycles than it would in reality, and any peak that builds over several reflections will be overstated.

Problem 11.2. A rational fit of $A(\omega )$ uses 12 poles. Roughly how much work per line end per step does the convolution cost, and how does it compare with evaluating the convolution directly for a line whose impulse response lasts $2\text{ms}$ at $\mathrm{\Delta }t=20\mu \text{s}$?

Solution 11.2

With recursive convolution each pole is one two-term update, so 12 poles cost on the order of 12 multiply-accumulates — call it a few dozen floating-point operations, and crucially a constant independent of the line.

Direct evaluation would need one product per stored history sample: $2\text{ms}/20\mu \text{s}=100$ samples, so about 100 operations, and that number grows with both the length of the response and the fineness of the step. Halving $\mathrm{\Delta }t$ doubles the direct cost while leaving the recursive cost unchanged.

The factor of eight here understates the case. Real responses on long lines last many milliseconds, and the memory traffic of walking a long history buffer costs more than the arithmetic. The recursive form is what makes the model viable at all.

Problem 11.3. Why can a finite-order rational function not represent $e^{-j\omega \tau }$, and what is done instead?

Solution 11.3

The phase of $e^{-j\omega \tau }$ is $-\omega \tau$, which decreases without bound as $\omega$ grows. A rational function of finite order has a phase that is a sum of finitely many arctangent terms, and each contributes at most $90°$, so the total phase it can express is bounded. No finite order can track an unbounded phase ramp; adding poles only extends the frequency range over which the failure is postponed.

The remedy is not to try. The delay is factored out as $A=e^{-j\omega {\tau }_{min}}{A}^{\prime }$, only the smooth remainder $A^{\prime }$ is fitted, and the delay is implemented separately as an integer-step buffer — the same mechanism Chapter 8 already uses. Splitting the problem this way lets each half be handled by the tool suited to it.

Problem 11.4. A study of a lightning strike on a substation entrance is requested, and the only line data available was computed at 60 Hz. What should the requester be told?

Solution 11.4

That the data is unfit for the study, together with the reason.

A lightning front has significant content up to the megahertz range. At those frequencies the earth-return resistance is orders of magnitude above its 60 Hz value, and the conductor's own resistance is elevated by skin effect as well. A constant-parameter model built on 60 Hz data will transport the front almost undistorted over tens of kilometres, whereas a real line rounds it off substantially. Since insulation coordination turns on the rate of rise and the peak of that front, the error is in the unsafe direction — the simulation will make the incoming surge look steeper than it will be.

The constructive answers, in order of preference: obtain or compute line data at a frequency representative of the phenomenon; use a frequency-dependent model if one is available; or restrict the study's conclusions to quantities that do not depend on front shape. What is not acceptable is running the 60 Hz data and reporting the front as computed.

Problem 11.5. Explain why phase-domain fitting is preferred for cables even though it requires fitting $n^2$ functions instead of $n$.

Solution 11.5

Because for cables the modal transformation matrix is strongly frequency-dependent, and approximating it by a real constant is the dominant error — larger than anything the modal fits themselves contribute. A cable's conductors sit inside a dielectric with a metallic sheath close by, so the coupling between conductors is far stronger and far more frequency-sensitive than on an overhead line, and the eigenvectors move correspondingly.

Phase-domain fitting removes that error entirely because there is no transformation matrix to approximate. The extra cost is real — $n^2$ fitted functions, and a harder passivity constraint on a matrix fit than on a scalar one — but it buys the removal of an error that cannot be reduced any other way.

Problem 11.6. In NumaSim today, a single set of constant parameters must be chosen for a line used in a study covering both a 60 Hz steady-state period and a switching event. Which should govern the choice?

Solution 11.6

The transient should govern, with the steady state checked separately.

The argument rests on what each choice costs. Parameters chosen at a few kilohertz render the 60 Hz steady state slightly wrong, mostly in the resistive drop — an error that can be quantified in advance and, where it matters, corrected by comparison against a load-flow calculation. Parameters chosen at 60 Hz make the transient's damping wrong by a factor of ten or more, which is neither a small correction nor easy to bound after the fact.

Where both parts of the answer must be accurate, the sound approach is two runs with two parameter sets, each used only for the portion of the answer for which it is valid, with the division stated in the report.

11.12 References

• J. R. Marti, "Accurate modelling of frequency-dependent transmission lines in electromagnetic transient simulations", IEEE Trans. PAS, 1982 — rational fitting of $A(\omega )$ and $Z_C(\omega )$ with the delay extracted.
• A. Semlyen and A. Dabuleanu, "Fast and accurate switching transient calculations on transmission lines with ground return using recursive convolutions", IEEE Trans. PAS, 1975 — the recursive convolution used in §11.5.
• B. Gustavsen and A. Semlyen, "Rational approximation of frequency domain responses by vector fitting", IEEE Trans. Power Delivery, 1999 — the fitting algorithm now used almost universally.
• J. Arrillaga and N. R. Watson, Power Systems Electromagnetic Transients Simulation, IET Power and Energy Series 39 — frequency-dependent line models, back-winding, and phase- versus modal-domain fitting.

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Previous: Chapter 10 — Multiconductor lines · Next: Chapter 12 — Line and cable parameters.