Corona and Radio Interference

Everything in Part 2 and Part 3 has treated the line as a set of circuit parameters \(R\), \(L\), \(C\) and \(G\), with the air around the conductors doing nothing but hold the charge in place. Chapter 2 argued that transmission voltage should be as high as the design will allow, because the power a line can carry rises as \(V^2\) while its \(I^2R\) loss falls as \(1/V^2\). Chapter 14 pushed the same argument to its conclusion with surge impedance loading. Nothing so far has said where the argument stops.

It stops at the conductor surface. Chapter 8 computed the quantity that matters, the surface voltage gradient of a single conductor carrying phase-to-neutral voltage \(V_{an}\):

Raise \(V_{an}\) and \(E_r\) rises in exact proportion. The strength of air, however, does not rise with anything. It is fixed at roughly \(30\) kilovolts per centimetre by the physics of nitrogen and oxygen molecules, and once the gradient at the conductor surface reaches that value the air immediately around the conductor stops being an insulator. It becomes a thin, luminous, crackling shell of ionised gas that consumes power, radiates radio noise, generates ozone and hisses audibly. That is corona.

The phenomenon is not flashover. A line in corona keeps operating; the discharge is confined to a sheath a few millimetres thick and never bridges the gap to the next phase or to the tower. This chapter is about why the discharge stops where it does, at what voltage it starts, what it costs, and what a designer does about it. The answer to the last question was given in Chapter 8 before its justification: split the phase into a bundle. Here we finally compute the threshold that made bundling unavoidable.

Air is never perfectly neutral. Cosmic rays, ultraviolet light and the natural radioactivity of the ground keep a few hundred ion pairs per cubic centimetre in existence at all times, continuously created and continuously recombining. Those stray free electrons are the seed of everything that follows.

Put a free electron in a field \(E\). It is accelerated, travels a mean free path, and collides with a neutral molecule. Because the molecule is some tens of thousands of times heavier, an elastic collision returns only a tiny fraction of the electron's kinetic energy — the electron therefore keeps most of what the field gave it and is accelerated again. Over a succession of free paths its energy climbs until, at a collision, it exceeds the ionisation energy of the molecule (about \(15.6\) eV for nitrogen, \(12.1\) eV for oxygen). That collision knocks an electron loose. Where there was one free electron there are now two, and a positive ion left behind.

Both electrons repeat the process. The population grows geometrically along the direction of the field, which is Townsend's description of an avalanche: if \(\alpha\) is the number of ionising collisions an electron makes per centimetre of drift, then a single electron starting at \(x=0\) has become

The coefficient \(\alpha\) depends on the field only through the ratio \(E/p\), because the field sets the energy gained per free path and the pressure sets the length of that free path. This single fact is the origin of the air density correction of Section 15-4, and it is worth carrying forward.

Now the crucial question. If an avalanche multiplies without limit, why does the line not simply flash over? Because the field around a cylindrical conductor is not uniform. At a radial distance \(x\) from the conductor axis the field is

and it falls as \(1/x\). Only in the region where \(E(x)\) exceeds the threshold can \(\alpha\) outrun the rate at which electrons are lost to attachment on oxygen molecules. Beyond that radius the avalanche starves. Denote the threshold gradient by \(g_o\) and find the radius at which the field has decayed to it:

where \(V_c = g_o\, r \ln(d/r)\) is the voltage at which the surface itself has just reached the threshold — the critical disruptive voltage derived properly in the next section. The result is exact and unusually informative: the ionised shell extends to \(r\) times the per-unit overvoltage. At \(V = 1.3\,V_c\) the glowing envelope reaches \(1.3r\); at twice the critical voltage it reaches \(2r\). The discharge is self-limiting because it must live inside a field that it cannot itself extend.

The same relation quietly explains a claim made in every list of corona's "advantages": that corona increases the effective diameter of the conductor. It does, and by exactly the factor \(V/V_c\), because the conducting plasma sheath behaves electrically as part of the conductor.

One asymmetry is worth recording because it shows up on a direct-current line. On a negative conductor the avalanche is launched from the conductor surface into a falling field and terminates quickly, producing a rapid, highly regular train of small pulses (Trichel pulses) and a spotty, mobile glow. On a positive conductor the electrons move toward the conductor, into a rising field, so the discharge develops as long filamentary streamers that push outward — larger pulses, less regular, and a smooth uniform glow. Under alternating voltage both occur, one per half cycle, and the positive streamers dominate everything that follows in Section 15-7.

The critical disruptive voltage \(V_c\) is defined as the lowest phase-to-neutral voltage at which the air at the conductor surface starts to ionise. Its derivation is now a single substitution.

Take two parallel conductors of radius \(r\) whose axes are \(d\) apart, with \(d \gg r\), and let \(V\) be the phase-to-neutral voltage. Chapter 7 gives the charge per unit length as \(q = 2\pi\varepsilon V/\ln(d/r)\), and the field at the surface of a line charge is \(q/2\pi\varepsilon r\), so the surface gradient is

Corona begins the instant \(g\) reaches the breakdown strength \(g_o\). Setting \(g = g_o\) and solving for the voltage that does it:

Two extensions make this usable. For a three-phase line the spacing \(d\) is replaced by the equivalent equilateral spacing — the geometric mean distance \(D_{eq} = \sqrt[3]{D_{12}D_{23}D_{31}}\) introduced in Chapter 6 and used again for capacitance in Chapter 7. The logarithm is the same logarithm that appeared in both of those chapters, which is not a coincidence: corona, inductance and capacitance are all statements about the same field pattern.

The second extension is that \(V_c\) as written applies to a mirror-smooth conductor in laboratory air. Real conductors are stranded and real air is not at \(76\) cm of mercury. Both corrections multiply the expression, and both reduce it.

Section 15-2 established that the ionisation coefficient depends on \(E/p\). Turn that around: the field required to sustain ionisation is proportional to the pressure, or more precisely to the number density of molecules, because what really matters is the energy an electron picks up between collisions and that is set by how far apart the molecules are. By the ideal gas law the number density is proportional to \(p/T\). Hence

The famous constant is nothing more than \(298/76 = 3.921\), with \(b\) the barometric pressure in centimetres of mercury and \(t\) the ambient temperature in degrees Celsius. At \(b = 76\) and \(t = 25^\circ\)C the factor is exactly unity, as it must be. Everywhere else it is less than one: hotter air is thinner, and higher ground has less air above it.

A line at \(2000\) m on a summer afternoon has lost a quarter of its corona margin relative to the same line at sea level in the morning. Substation clearances and insulator creepage in mountainous regions are derated for the same reason.

The second correction is geometric rather than atmospheric. The derivation of \(V_c\) assumed a perfectly cylindrical, perfectly smooth conductor, so that the field on the surface is the same all the way round. A stranded conductor is scalloped: each outer strand is a small cylinder of its own, and the field at the crown of a strand is appreciably higher than the field a smooth cylinder of the same overall diameter would produce. Dirt, scratches from stringing, insect bodies and water droplets do the same thing locally and much more violently. All of it is lumped into an empirical irregularity factor \(m_o \le 1\) multiplying \(V_c\).

The equivalent statement in terms of gradient is often more convenient in design: corona begins when the computed rms surface gradient reaches \(m_o g_o \delta\). For a stranded conductor at sea level that threshold is about \(0.85 \times 21.1 = 17.9\) kV/cm, which is why Chapter 8 quoted a design target of \(15\) to \(17\) kV/cm — a margin of roughly ten to fifteen per cent against fair-weather corona, and no margin at all against rain.

Raise a line slowly past \(V_c\) at night and nothing is seen. Ionisation is occurring — a sensitive instrument registers loss and radio noise — but the eye detects nothing until the voltage is appreciably higher. The voltage at which a continuous glow appears along the whole conductor is the visual critical voltage \(V_v\), and it always exceeds \(V_c\).

The reason is the one already established in Section 15-2. Visible light comes from excited molecules relaxing, and enough of them must be excited within a shell thick enough to register. At \(V\) barely above \(V_c\) the envelope radius \(x_0 = r\,V/V_c\) is barely above \(r\): the ionised shell is a few tens of micrometres thick and produces no perceptible light. The shell must reach an appreciable absolute thickness before it glows, and on a thin conductor, where the \(1/x\) field collapses quickly, that requires a much larger overvoltage than on a fat one.

Peek measured the effect and fitted it. The extra thickness needed turns out to scale as \(\sqrt{\delta r}\), so the correction enters as a term \(0.3/\sqrt{\delta r}\) added to unity:

The bracket alone shows how the effect fades with size. At \(\delta = 1\) it is \(1.42\) for a \(0.5\) cm conductor, \(1.30\) at \(1\) cm, \(1.21\) at \(2\) cm and \(1.17\) at \(3\) cm: on a very thin wire the glow is delayed by more than \(40\%\) in voltage, on a heavy transmission conductor by less than \(20\%\). Because \(m_v\) is well below \(m_o\) for stranded conductors, the two effects partly cancel, and the ratio \(V_v/V_c\) for a real stranded line is typically only \(1.05\) to \(1.15\).

Every ionising collision costs energy, and the energy comes from the line. It leaves as light, heat, sound, and the chemical energy locked up in ozone and oxides of nitrogen. Before quoting the standard formula it is worth seeing why it has the shape it has.

Consider one half cycle. As the instantaneous voltage climbs past \(V_c\), streamers form and inject a space charge into the sheath. That charge is proportional to how far the voltage has exceeded the critical value, so \(q_{corona} \propto (V - V_c)\). Sweeping it out through the potential difference that drove it costs energy \(\propto (V-V_c) \times q_{corona} \propto (V-V_c)^2\). This happens twice per cycle, so the power is proportional to frequency:

Peek's experiments on test lines supplied the constant and a weak geometric term \(\sqrt{r/d}\), which accounts for how quickly the field falls away from conductors of different slenderness. He also found that the loss did not extrapolate to zero at zero frequency — a direct-voltage conductor in corona still loses power — which is why the frequency appears as \(f + 25\) rather than \(f\).

The formula is empirical and its range of validity is narrow. It should be applied only when the supply frequency lies between \(25\) and \(120\) Hz, when the ratio of operating to critical voltage is above about \(0.8\) — that is, when the line is anywhere near the threshold — and when the conductor radius exceeds \(0.25\) cm. Outside those bounds, and particularly for \(V\) very close to \(V_c\), Peek's expression overestimates the loss, sometimes by a large factor, because the quadratic law is a fit to the well-developed discharge and not to its onset.

For the region close to and below the threshold, Peterson's formula is preferred. It replaces the quadratic in \((V-V_c)\) with a term in \(V^2\) multiplied by an experimentally determined function \(F\) of the ratio \(V/V_c\):

Here \(F\) is read from a measured curve; it is very small below \(V/V_c \approx 0.8\), passes through a knee near unity, and climbs steeply beyond it. The form is instructive even without the curve: the loss is governed by the ratio \(V/V_c\), so a line designed with a comfortable margin loses essentially nothing and a line designed on the margin loses a great deal.

Rain, snow, fog and hoar frost transform the picture. Water hanging from the underside of a conductor forms drops that are drawn into sharp points by the field, and each point is a local field concentrator far worse than any stranding. The universal engineering allowance is to take the critical disruptive voltage in foul weather as

Because the loss is quadratic in the excess, a \(20\%\) reduction in threshold does far more than a \(20\%\) increase in loss. On a line operating modestly above \(V_c\) the excess \((V-V_c)\) can triple, and the loss therefore rises by nearly an order of magnitude — which is exactly what Example 3 finds. Annual corona energy on a well-designed extra-high-voltage line is dominated almost entirely by the small fraction of hours in which it is raining.

Corona loss is a slow, averaged quantity. The discharge that produces it is anything but slow. Each streamer is a current pulse of a few tens of milliamperes lasting on the order of a hundred nanoseconds, and a conductor in corona emits a great many of them per half cycle at randomly scattered points along its length.

A pulse that short has a very wide spectrum. When it injects charge onto the conductor, the resulting current step splits and travels both ways along the line as a travelling wave at the surge impedance of Chapter 12, and the line — a horizontal wire kilometres long, a few tens of metres above a conducting plane — radiates. That radiation is radio interference, and it lands squarely in the medium-wave broadcast band.

The two polarities behave differently, and the difference matters. Negative Trichel pulses are small, short and extremely regular; positive streamer pulses are perhaps an order of magnitude larger in amplitude and much less regular. On an ac line, radio interference is therefore produced almost entirely during the positive half cycle of each phase.

Because the mechanism is governed by the surface gradient, so is the interference. A widely used empirical expression, fitted to fair-weather measurements at \(1\) MHz, is

with \(E_m\) the maximum conductor surface gradient in kV/cm rms, \(d_c\) the conductor (or sub-conductor) diameter in cm, and \(D\) the radial distance from the conductor in metres. Three features of it deserve attention.

First, the gradient enters linearly in decibels, so every extra kV/cm of surface gradient costs \(3.5\) dB — a factor of \(1.5\) in field strength. Interference is an exponential function of gradient, and this is why a design that is marginal on corona loss is hopeless on interference.

Second, the lateral decay is \(33\log_{10}D\), roughly \(10\) dB per doubling of distance, which is faster than the \(6\) dB per doubling of a single radiating wire. The extra attenuation comes from cancellation between the three phases, whose corona sources are in different places and at different instants of the cycle.

Third, the frequency dependence is steep: above \(1\) MHz the level falls by \(20\) to \(30\) dB per decade. By the time one reaches the VHF television band the contribution of conductor corona is negligible. Television interference near a line is almost always traced to gap-type discharges — a loose split pin, a corroded joint in an insulator string, a cracked ceramic — and not to the conductors themselves, which is a useful diagnostic rule in the field.

What matters to the neighbour is not the noise but the ratio of the wanted signal to it. The usual acceptability criterion for amplitude-modulated broadcast reception is a signal-to-noise ratio of about \(24\) dB at the edge of the right of way, and typical utility specifications limit fair-weather interference to \(40\) to \(45\) dB(\(\mu\)V/m) there. Rain raises the measured level by \(15\) to \(25\) dB, so foul-weather interference is accepted rather than designed against — on the reasoning that atmospheric noise from the same storm is high anyway.

Alongside the radio noise there is sound. A conductor in corona emits a broadband hiss and crackle from the individual streamers, plus a pure tone at twice power frequency produced by the space charge oscillating in and out of the sheath each half cycle. Audible noise is quoted in A-weighted decibels and, like interference, rises very steeply with surface gradient. It is negligible in fair weather and prominent in rain. Design limits at the edge of the right of way are of the order of \(52\) to \(55\) dBA in rain.

Corona is usually presented as a defect, and mostly it is. It is worth being precise about the two respects in which it genuinely helps, because both follow from results already derived and both matter to later chapters.

The first benefit is the enlarged effective radius. Section 15-2 showed that the conducting sheath extends to \(x_0 = r\,V/V_c\). A conducting shell at the conductor potential is the conductor, electrically, so the gradient at the outer edge of the sheath is lower than it would be at a bare surface of radius \(r\). Corona partly relieves the very stress that created it, which is why the discharge stabilises instead of running away into a flashover.

The second benefit is the effect on travelling waves. A lightning or switching surge arriving on a line has a crest far above the operating voltage, so it drives the conductor deep into corona for the duration of its front. The discharge draws energy out of the wave, and the extra charge stored in the sheath acts as additional shunt capacitance which slows the crest of the wave relative to its toe. The surge therefore arrives at the substation both attenuated and with a gentler front than it left with, which relaxes the insulation coordination problem of Chapter 37 considerably. Corona is, in this sense, a distributed surge absorber that costs nothing to install.

Against that stand the costs.

Set against the benefits, the balance is decisively negative for steady-state operation and mildly positive for surges. The engineering conclusion is therefore not "eliminate corona" but "keep the line out of corona at operating voltage, and be glad of the corona that appears when a surge passes".

Every remedy is an attempt to satisfy one inequality:

Only the left-hand side is under the designer's control, and Chapter 8 already catalogued what can be done to it. Ranked by how much they actually buy:

Increase the conductor radius. Since \(E \propto 1/[r\ln(D_{eq}/r)]\), this is the strongest single lever, and it is the reason ACSR exists: the steel core lets a conductor have a large diameter without the aluminium cross-section — and therefore the cost and weight — that a solid conductor of the same diameter would need. Where an even larger diameter was wanted without more metal, expanded ACSR with a fibre or paper filler between core and outer layers was used.

Split the phase into a bundle. The gradient of an \(n\)-conductor bundle carries the factor \(n\) in its denominator, as Chapter 8 derived, and the equivalent bundle radius \(r_b\) inside the logarithm helps a little more. This is the only lever that scales, and above about \(220\) kV no line is built without it.

Widen the spacing. \(D_{eq}\) sits inside a logarithm, so the return is poor — doubling the spacing improves the gradient by perhaps \(13\%\) — while tower cross-arms, right-of-way width and line inductance all grow with it. Spacing is set by clearance and swing requirements, not by corona.

Control the surface and the hardware. The factor \(m_o\) is not a constant of nature; it is a description of how carefully the line was built and maintained. Conductors are strung over rollers to avoid scoring, joints are compressed and smoothed, and every fitting that would otherwise present an edge is shrouded. Corona rings — toroids fitted at the line end of an insulator string and around the terminals of high-voltage bushings, arresters and instrument transformers — exist purely to replace a sharp geometry with a large radius of curvature and so cap the local gradient. Inside substations the same reasoning applies to bus-bar corners and clamp designs at \(33\) kV and above.

The last column is the point of the table. Across a factor of nine in system voltage the surface gradient barely moves. That constancy is not a coincidence or a convention — it is the design condition above, enforced at every voltage level, with the bundle count doing the work that the voltage would otherwise force onto the air.

Examples 1 to 6 follow a single line so that each result can be checked against the last: a three-phase \(220\) kV, \(50\) Hz overhead line with conductors of radius \(1.2\) cm arranged at an equivalent spacing of \(250\) cm, with \(m_o = 0.85\) and \(m_v = 0.72\).

Take \(g_o = 21.1\) kV/cm rms unless a problem states otherwise, and be explicit about whether each voltage you compute is per phase or line to line — most marks lost on corona questions are lost to that one confusion.