Capacitive Switching

Unloaded Cable Switching: Charging Current and Inrush

Switching an unloaded high-voltage cable — how the cable’s shunt capacitance sets the charging current on de-energising, how the recovery voltage follows the screened or belted construction, and how the energising inrush is bounded for a single cable and for the more onerous back-to-back case where the small inter-cable inductance drives the rate of change. Follows IEEE Std C37.012-2022, with APS engineering interpretation.

Reading time ≈ 19 min

Section 1

Unloaded Cables as a Capacitive Switching Duty

An unloaded cable behaves, electrically, mainly as a capacitance. Because the conductor and the earthed screen are separated only by a thin solid dielectric, a power cable stores far more charge per kilometre than an equivalent overhead line, so even a moderate cable length can produce a significant charging current. The important practical point is that cable length becomes limiting much earlier than overhead-line length, because the capacitance per kilometre is so much higher. When the cable is opened, the breaker interrupts this capacitive current and a recovery voltage appears across the open contacts; when it is closed, the same stored-energy behaviour drives the energising inrush current. Switching an open-ended cable is one of the four recognised capacitive-current duties for high-voltage circuit breakers.

This note follows the treatment in IEEE Std C37.012-2022. This page develops the unloaded-cable duty introduced in the general application overview, focusing on the cable charging current, the recovery voltage after de-energising, and single-cable versus back-to-back energising inrush. For the capacitor-bank analogue see energising capacitor banks: inrush and back-to-back switching.

The single-phase equivalent circuit

The two-cable representation is used so that both duties can be described with one equivalent circuit: switching one cable alone, or switching one cable while another is already energised on the same bus. Two cables are taken to share a bus, each supplied through its own circuit breaker. Cable 1 is represented by its capacitance C₁ and surge impedance Z₁, cable 2 by C₂ and Z₂. Between them sit the source inductance L_s, the bus inductances L_b1 and L_b2, and the connection inductances L₁ and L₂ from each cable to the bus. Whether the duty is single or back-to-back depends only on whether cable 2 is in service when cable 1 is switched.

The two switching modes stress the breaker very differently, so it is worth separating them at the outset. In single-cable switching the source and cable surge impedance limit the inrush current. In back-to-back switching an already-energised cable discharges into the cable being switched through a short, low-inductance path, which makes the current far steeper and usually more onerous for the breaker.

Table 1 — The two cable-switching modes.

Mode	Circuit Condition	Governing Concern
Single cable	Cable 1 switched with cable 2 not connected; the current path is the source inductance in series with the bus and connection inductances.	Inrush peak is limited by the cable surge impedance and stays below the available short-circuit current, so it rarely governs.
Back-to-back	Cable 1 switched while cable 2 is already energised on the same bus; the source is effectively bypassed by the small inter-cable inductance.	Very high initial rate of change and equivalent frequency of the inter-cable surge current — the parameters that stress the making device.

Screened and belted cables

Three-phase cables come in two constructions that behave differently on interruption. A screened cable gives every core its own earthed metallic screen, so each phase is an independent conductor-to-earth capacitance. A belted cable places a common belt of insulation and a single screen around all three cores, so phase-to-phase capacitances couple the conductors. That difference decides the shape of the recovery voltage after the current is interrupted (Section 3). In practice this matters because a screened cable presents mainly a phase-to-earth recovery-voltage duty, like an earthed-neutral capacitor bank, while a belted cable’s stronger phase-to-phase coupling raises the recovery voltage towards the unearthed-bank and line-like cases treated elsewhere in the series.

Section 2

De-energising: the Cable Charging Current

The current a breaker interrupts when de-energising an unloaded cable is the cable’s charging current — the capacitive current drawn by the shunt capacitance at operating voltage. It is not a load current; it is the capacitive current needed to charge and discharge the cable insulation each power-frequency cycle. It is fixed by the system voltage, the cable geometry, the dielectric constant of the insulation, and the length. The shunt capacitive reactance can be taken from the manufacturer’s data or, if the physical dimensions are known, calculated from the cable cross-section.

\[ C = \frac{2\pi\,\varepsilon_{0}\,\varepsilon_{r}}{\ln\!\left(r_{i}/r_{c}\right)} \qquad\qquad X_{c} = \frac{6.58}{f_{s}\,\varepsilon_{r}}\,\log_{10}\!\frac{r_{i}}{r_{c}} \]

\(C\): shunt capacitance per unit length (F/m)
\(X_{c}\): shunt capacitive reactance (MΩ per phase per km)
\(\varepsilon_{0}\): permittivity of free space, \(8.85\times10^{-12}\) F/m
\(\varepsilon_{r}\): relative permittivity of the dielectric
\(f_{s}\): system frequency (Hz)
\(r_{i}\): inside radius of the insulation shield (mm)
\(r_{c}\): radius of the conductor shield — about 0.7 mm larger than the conductor for extruded insulation (mm)

The reactance expression is written per kilometre. Because cable capacitance adds with length, the total capacitance increases as the cable gets longer, so the equivalent capacitive reactance decreases with length and the charging current increases with length. In practice the per-kilometre value from \(X_{c}\) above is divided by the number of kilometres of cable before the charging current is found. The per-phase charging current then follows as \(I_{c} = U/(\sqrt{3}\,X_{c,\text{total}})\), and this is compared with the breaker’s rated cable charging current from IEEE Std C37.04; if the duty exceeds the rating the manufacturer should be consulted, and the inrush rating (Sections 4–5) must also be checked before the application is confirmed.

Table 2 — Typical relative permittivity of cable dielectrics.

Dielectric	Relative Permittivity \(\varepsilon_{r}\)
Gas-insulated	about 1.02
Polyethylene / XLPE	about 2.3
Paper–polypropylene–paper (PPP)	about 3.0
Fluid-impregnated paper	up to 4.0

Why cables cap the practical circuit length

For the same voltage a cable draws roughly twenty to forty times the charging current of an overhead line, because the conductor-to-screen spacing is small and the dielectric constant is several times that of air. A modest cable length can therefore approach the breaker’s rated capacitive current, and the accumulated charging current is one of the reasons long EHV cable circuits need shunt reactor compensation. This is also why the cable charging-current check should be made before assuming that an existing line-or-cable breaker rating is adequate for a long cable circuit.

Section 3

De-energising: the Recovery Voltage

After interruption the cable does not immediately lose its charge. The cable-side terminal can stay charged while the source-side terminal continues to follow the system voltage, so the voltage across the open breaker contacts is the difference between the two — and that difference is the recovery voltage. Its shape depends on the cable construction, and each construction maps neatly onto a duty already characterised elsewhere: the two differ because a screened cable acts as three separate phase-to-earth capacitances, while a belted cable’s common belt couples the phases.

Table 3 — Recovery-voltage analogue by cable construction.

Construction	Capacitive Behaviour	Recovery Voltage Behaves Like
Screened cable	Each core has its own earthed screen; phases do not interact capacitively.	An earthed-neutral capacitor bank — the recovery voltage is that of the solidly earthed case.
Belted cable	A common belt and screen couple the three cores through phase-to-phase capacitance.	An uncompensated transmission line — the interacting phases raise the recovery voltage above the earthed case.

The practical consequence is that a breaker switching belted cable must be assessed against the higher recovery voltage of the unearthed / line-like case, exactly as a capacitor-bank breaker on an unearthed bank is held to a larger voltage factor. Screened cable, which dominates modern HV installations, sits at the milder earthed-neutral end — its recovery voltage is that of a solidly earthed capacitor bank, about 2.0 pu, unless the run exceeds roughly 100 km, where the Ferranti effect begins to lift it. Belted cable is now rare and almost always below 38 kV.

The low cable surge impedance has a second consequence on energising: at roughly 20 Ω to 30 Ω it is an order of magnitude below an overhead line’s 300–450 Ω, so a cable’s energising inrush is about ten times that of an equivalent line — one more reason back-to-back cable switching (Section 6) demands care.

Section 4

Energising an Unloaded Cable

Closing a breaker onto an unloaded cable produces a transient inrush current as the shunt capacitance charges. Before energising, the cable is normally at earth potential but may carry a trapped charge left by a previous switching operation, and that residual voltage adds directly to the driving voltage. The magnitude and rate of rise of the inrush depend on:

the driving voltage, including the point on the voltage wave at the instant of closing;
the cable surge impedance and its capacitive reactance;
the amount and location of inductance in the circuit;
any trapped charge on the cable when the contacts touch;
any damping from closing resistors or other circuit resistance.

For a single cable the inrush peak is smaller than the available short-circuit current at the breaker terminals, so — since the breaker must already meet the system’s making-current requirement — single-cable inrush is never the limiting factor. The difficulty is confined to back-to-back switching, where a high-magnitude, high-frequency surge flows between cables through only the small inter-cable inductance, on closing or on a restrike during opening. That surge decays to zero within a fraction of a power-frequency cycle, and the component supplied by the source is so slow by comparison that it can be neglected.

Section 5

Energising a Single Cable

When a single unloaded cable is energised, the source supplies the initial charging current into the cable capacitance. That current is limited by the source, bus and connection inductances together with the cable surge impedance, so the peak is normally bounded by the available short-circuit current. More formally, a cable is classed as single when the maximum rate of change of its inrush current, on energising an uncharged cable, does not exceed the rate of change associated with the maximum symmetrical interrupting current. That limiting slope is the same threshold used for capacitor banks:

The single / back-to-back threshold

\( \left(di/dt\right)_{\max} = \sqrt{2}\,\omega_{s}\,I_{sc} = 2\sqrt{2}\,\pi f_{s}\,I_{sc} \), with \(I_{sc}\) the rated rms short-circuit current and \(\omega_{s}=2\pi f_{s}\). A circuit that is physically back-to-back can still be treated as single for application purposes if a large enough inductance sits between the two cables — large enough that, on its own, it would limit the fault current to at most the breaker rating.

How the cable is represented depends on the surrounding inductance. If the source inductance exceeds about ten times the cable inductance, the cable behaves as a lumped capacitor; otherwise it must be treated under transient conditions by its surge impedance:

\[ Z = \sqrt{\frac{L}{C}} = \frac{138}{\sqrt{\varepsilon_{r}}}\,\log_{10}\!\frac{r_{i}}{r_{c}} \]

\(Z\): cable surge impedance (Ω); typically 20–50 Ω, with 50 Ω a common value
\(L\): distributed series inductance per unit length, \(L = (\mu_{0}\mu_{r}/2\pi)\ln(r_{i}/r_{c})\) (H/m)
\(C\): distributed shunt capacitance per unit length (F/m)
\(\varepsilon_{r}\): relative permittivity, 2.3 (polyethylene) to 4.0 (impregnated paper)

With the source contribution taken through the total inductance \(L = L_{s}+L_{b1}+L_{1}\) between source and cable, the inrush is a single damped exponential rather than an oscillation:

\[ i_{i}(t) = \frac{u_{m}-u_{t}}{Z_{1}}\,e^{-\frac{Z_{1}}{L}\,t}, \qquad \hat{\imath}_{i} = \frac{u_{m}-u_{t}}{Z_{1}}, \qquad \left.\frac{di_{i}}{dt}\right|_{t=0} = \frac{u_{m}-u_{t}}{L} \]

\(u_{m}\): crest of the applied voltage (V)
\(u_{t}\): trapped voltage on the cable being switched (V)
\(\hat{\imath}_{i}\): peak of the inrush current (A)
\(Z_{1}\): surge impedance of the switched cable (Ω)
\(L\): inductance between the source and the cable (H)

The inrush is not oscillatory in the usual sense, but its initial slope defines an equivalent frequency that can be checked against the breaker’s rated inrush frequency. Matching the initial slope to a sinusoid peaking at the rated inrush current gives \( f_{eq} = (u_{m}-u_{t})/(2\pi L\,\hat{\imath}_{ir}) \); for a valid application \(f_{eq}\) should stay below four times the tested inrush-current frequency.

Worked check — a 132 kV single cable

This check contrasts the relatively modest charging-current duty with the potentially more onerous inrush duty. Take a screened 132 kV cable, surge impedance \(Z_{1}=30\) Ω, energised with no trapped charge. The crest phase-to-earth voltage is \(u_{m}=132\sqrt{2}/\sqrt{3}\approx108\) kV, so the peak inrush is \(\hat{\imath}_{i}=108{,}000/30\approx\mathbf{3.6\ kA}\). A typical three-phase short-circuit current at 132 kV is tens of kiloamperes, so the single-cable inrush is comfortably below it — confirming that single-cable energising is not the governing duty. The result should still be checked against both the breaker’s rated cable charging current and its making/inrush capability.

Section 6

Back-to-Back Cable Energising

In back-to-back cable switching the already-energised cable behaves like a charged energy source: the incoming cable is charged through the short interconnection between the two cable circuits, so the source impedance is partly bypassed, and the result can be a very steep current with a high equivalent frequency. Cables are switched back-to-back when the rate of change of the inrush exceeds the single-cable threshold — the case where cable 2 is already in service as cable 1 is energised. The inductances L₁, L₂, L_b1 and L_b2 between the cables are usually a small fraction of the source inductance, often below 1%: they are simply the connections from cables to breakers, the breaker inductances and the bus. Their value is site-specific and cannot be standardised, but a representative range is 0.66 µH to 1.0 µH per phase per metre. With the source contribution neglected, the circuit reduces to the switched cable and the already-connected cable in series through the total loop inductance:

\[ \hat{\imath}_{i} = \frac{u_{m}-u_{t}}{Z_{1}+Z_{2}} \;\;(\text{unequal cables}) \qquad \hat{\imath}_{i} = \frac{u_{m}-u_{t}}{2Z} \;\;(\text{equal cables}) \qquad \left.\frac{di_{i}}{dt}\right|_{t=0} = \frac{u_{m}-u_{t}}{L} \]

\(Z_{1},Z_{2}\): surge impedances of the switched and the already-energised cable (Ω)
\(Z\): common surge impedance when the two cables are equal (Ω)
\(L\): total inter-cable inductance, \(L = L_{1}+L_{b1}+L_{b2}+L_{2}\) (H)
\(\tau\): front time constant, \(\tau = L/(Z_{1}+Z_{2})\) (s)

Two features distinguish this case from the single cable. First, the surge impedances still limit the peak, so — unlike a back-to-back capacitor bank — the peak inrush of a back-to-back cable is modest, and in the equal-cable case it is actually lower than the single-cable peak. Second, and decisively, the rate of change is governed by the tiny inter-cable inductance instead of the source inductance, so the initial \(di/dt\) and the equivalent frequency can be extreme. It is the frequency and the \(i\times f\) product, not the peak amplitude, that stress the making device. The equivalent frequency matters because the breaker contacts, the pre-arcing process and nearby equipment are stressed not only by the current magnitude but by how fast the current rises.

Key idea

Adding cable to the bus does not raise the back-to-back peak — the surge impedance caps it — but it does collapse the loop inductance, which drives the equivalent frequency up. The design lever is therefore series inductance: a transient-limiting inductor deliberately added to L raises \(\tau\), lowers the initial slope, and brings \(f_{eq}\) back below four times the tested inrush frequency. In practice the case is quantified with an EMT study — for example an EMTP®-type simulation — because the travelling-wave reflections and the exact loop inductance decide the result.

Worked check — a second equal cable on the bus

Energising the same 132 kV cable (\(u_{m}\approx108\) kV, \(Z=30\) Ω) against an equal energised cable gives a peak inrush of only \(\hat{\imath}_{i}=108{,}000/60\approx\mathbf{1.8\ kA}\) — below the single-cable figure. But with just \(L\approx8\) µH of loop inductance (roughly 0.8 µH/m over ~10 m of bus) the initial slope is \((u_{m}-u_{t})/L = 108{,}000/8\times10^{-6}\approx\mathbf{13\ kA/\mu s}\); at \(L\approx24\) µH it is still about 4.5 kA/µs. Slopes of this order push the equivalent frequency far above the tested value, which is precisely why a series inductor is added.

Section 7

Inrush-Current Summary

Collecting the three configurations into one table gives the peak inrush current and the equivalent frequency to compare against the breaker rating. All use the surge-impedance representation; the symbols are those of Sections 5 and 6.

Table 4 — Inrush current and equivalent frequency for switching cables.

Configuration	Peak Inrush Current \(\hat{\imath}_{i}\)	Equivalent Frequency \(f_{eq}\)
Energising a single cable	\(\dfrac{u_{m}-u_{t}}{Z}\)	\(\dfrac{u_{m}-u_{t}}{2\pi L\,\hat{\imath}_{ir}}\)
Energising against another cable on the bus	\(\dfrac{u_{m}-u_{t}}{Z_{1}+Z_{2}}\)	\(\dfrac{u_{m}-u_{t}}{2\pi L\,\hat{\imath}_{ir}}\)
Energising against an equal cable on the bus	\(\dfrac{u_{m}-u_{t}}{2Z}\)	\(\dfrac{u_{m}-u_{t}}{2\pi L\,\hat{\imath}_{ir}}\)

For application, the peak inrush is checked against the rated value in IEEE Std C37.04, and the equivalent frequency against the tested inrush frequency. In extreme cases the high inrush is tolerable provided the back-to-back current \(i_{bb}\) stays below its rated value: the \(i\times f\) product is then acceptable up to four times the tested value for vacuum, SF₆ and other gas switchgear.

Section 8

Mixed and Alternate Configurations

Real substations rarely present a clean single or back-to-back cable. A cable often exits the station and joins an overhead line after a short run, or shares a bus with capacitor banks and other cables. When an overhead line and a cable are connected in the same switched circuit, the cable usually dominates the charging current because its capacitance per kilometre is so much higher, though the overhead line can still matter for travelling-wave behaviour, reflections and remote overvoltages. Other combinations of circuit elements can produce inrush currents with the same character as cable switching, and for application the peak inrush should always be checked against the breaker’s rated value.

A workable approach for a mixed circuit is to compare the relative contributions of its parts. For a short cable feeding a long line, the arrangement can be treated as a line with a capacitor to earth standing in for the cable; similar simplifications cover other combinations. The propagation speed inside the cable — \(v = 300{,}000/\sqrt{\varepsilon_{r}}\) km/s, so roughly 200,000 km/s for polyethylene — sets the travel time that decides whether the surge front behaves as a lumped pulse or a distributed travelling wave.

Section 9

Common Mistakes

The recurring errors in cable-switching applications come from treating a cable like an overhead line, confusing the steady charging current with the energising inrush, or carrying capacitor-bank intuition across to a surge-impedance-limited circuit.

Common mistakes to avoid

Judging the back-to-back duty by peak current — which the surge impedance keeps modest — and missing the very high \(di/dt\) and equivalent frequency that actually govern.
Using the charging current per kilometre without dividing by the number of kilometres, so a long cable’s current is badly underestimated.
Energising onto a trapped charge \(u_{t}\) of the opposite polarity and forgetting that it adds to \(u_{m}\), raising both the inrush peak and its rate of rise.
Treating a belted cable like a screened one and applying the earthed-neutral recovery voltage instead of the higher line-like value.
Assuming a physically back-to-back circuit is onerous when a large series inductance between cables makes it single for application — or the reverse, ignoring how little inductance is really there.
Adding a transient-limiting inductor to fix the frequency without rechecking the recovery voltage it imposes on the breaker.

Section 10

Key Points

Application checklist

Start with the cable charging-current check: cable capacitance per kilometre is high — twenty to forty times an overhead line — so the breaker’s rated cable charging current can become limiting even when the steady-state current appears modest.
Find the charging current from \(X_{c}=6.58\,\log_{10}(r_{i}/r_{c})/(f_{s}\varepsilon_{r})\) MΩ/phase/km, dividing by the cable length in km, and check it against the rated cable charging current.
Match the recovery voltage to the construction: screened cable behaves like an earthed-neutral bank, belted cable like an uncompensated line.
Treat single-cable inrush as non-governing — its surge-impedance-limited peak \((u_{m}-u_{t})/Z_{1}\) stays below the available short-circuit current.
Check whether another cable or capacitive circuit can be energised on the same bus; if so, assess the back-to-back duty, not only the single-cable one. Then watch the rate of change and equivalent frequency, not the peak: the small inter-cable inductance (0.66–1.0 µH/m) drives \(di/dt\) and \(f_{eq}\) up.
Keep \(f_{eq}\) below four times the tested inrush frequency, and the \(i\times f\) product within four times the tested value, adding series inductance where needed.
Quantify mixed cable-and-line circuits by comparing element contributions, and confirm the final peak inrush against the breaker rating — ideally through an EMT study.

Section 11

References and Further Reading

The standards and technical brochures behind this guide.

IEEE Std C37.012-2022, IEEE Guide for the Application of Capacitive Current Switching for AC High-Voltage Circuit Breakers Above 1000 V. New York, NY, USA: IEEE, 2022.
IEEE Std C37.04-2018, IEEE Standard for Ratings and Requirements for AC High-Voltage Circuit Breakers with Rated Maximum Voltage Above 1000 V. New York, NY, USA: IEEE, 2018.
IEEE Std C37.09-2018, IEEE Standard Test Procedures for AC High-Voltage Circuit Breakers with Rated Maximum Voltage Above 1000 V. New York, NY, USA: IEEE, 2018.
R. A. Gabrielle, P. G. Marchenko and G. S. Vassell, “Electrical constants and relative capacities of bundled-conductor transmission lines,” IEEE Trans. Power Apparatus and Systems, vol. 83, no. 7, pp. 743–751, 1964.
CIGRE, Shunt Capacitor Switching in Distribution and Transmission Systems, Technical Brochure 817. Paris, France: CIGRE, 2020.
J. C. Das, Transients in Electrical Systems: Analysis, Recognition, and Mitigation. New York, NY, USA: McGraw-Hill, 2010.

Part 3 Reading now

Unloaded Cable Switching

Cable charging current and capacitive reactance, screened versus belted recovery voltage, and single versus back-to-back cable inrush limited by the surge impedance.

Series progress 3 of 8

Part 4

Switching Unloaded Transmission Lines

Line charging current and the Ferranti effect, the C₁/C₀ recovery voltage, shunt compensation, energisation surges and the delayed-current-zero phenomenon.