Cálculos de diseño de anillo de corona: un ejemplo resuelto

When a commissioning crew first energizes a 220 kV line section, the hardware at the line end either stays quiet or announces itself with a hiss. That hiss is almost never a mystery. It is a sizing calculation that was skipped, rushed, or copied from the wrong project. This article walks through one complete corona ring design calculation for a 220 kV insulator string, from the specification sheet to the laboratory certificate, with every number shown.

The perspective here comes from the factory floor rather than the lecture hall. RaxPower manufactures pole line and overhead line hardware, and the factory floor behind it ships against the same acceptance logic this walkthrough follows. The walkthrough below applies the same discipline. State the inputs, show each step, and let measurement settle the argument.

Design Inputs: Reading the Specification Sheet First

A calculation is only as honest as its inputs, so the worked example starts from a written duty description. Picture a tension station on a new 220 kV three-phase line. The string is a composite longrod with a compression dead-end eye at the energized end, and the site is a coastal plain at sea level. A mountain variant at 2,000 m enters later in the walkthrough.

Five inputs drive the geometry, and the rest of the sheet is context. The first three set the electrical size, and the last two set the mechanical envelope.

• System voltage: 220 kV between phases. On a grounded-wye system the phase-to-ground value is 220 / √3 ≈ 127 kV RMS, whose crest is 127 × √2 ≈ 180 kV.
• Insulation arrangement: a composite longrod string whose line-end fitting concentrates the field where the conductor hardware begins.
• Site altitude: sea level for the base case, 2,000 m for the variant.
• Environment: humid coastal air, which argues for margin rather than a bare pass.
• Mechanical duty: span tension and short-circuit forces on the ring bracket, carried by the tube section and the yoke.

Fixing the Field Limit Before Any Geometry

Corona ring design starts with a field limit, not a diameter table. Air begins to ionize at a surface gradient in the class of 30 kV/cm under standard conditions. The classical Peek relation refines that threshold for curvature. It reads g_v = g₀ × δ × (1 + 0.301 / √(δ·r)), with g₀ taken as 30 kV/cm. Here δ is the relative air density and r the tube radius in centimetres.

A real surface is never the polished probe of a laboratory curve. Weathered aluminum carries an irregularity factor below unity, and the Peek framework puts roughened or dirty surfaces between 0.93 and 0.98. This example adopts m = 0.95. The working ceiling is then set at 80 percent of m × g_v, a margin that absorbs rain, pollution, and the edges of adjacent hardware.

Design ceiling = 0.8 × m × g_v. The 0.8 factor is a stated policy of this example, not a universal constant; a drier inland site might justify less, a polluted one more.

The two assumptions turn into numbers immediately. For a trial tube radius near 1 cm at sea level, the onset gradient works out to 30 × (1 + 0.301/1) ≈ 39 kV/cm. The ceiling follows at 0.8 × 0.95 × 39 ≈ 29.7 kV/cm, and everything that follows is checked against that ceiling.

First-Pass Tube Diameter From the Onset Gradient

The first pass of a corona ring design calculation is a screening estimate, and it should be quick. Treat the ring as an isolated thin toroid at the crest potential. Start from the standard thin-torus capacitance approximation, C ≈ 2π²ε₀R / ln(8R/r). It yields a surface field estimate of E ≈ Û / (2r·ln(8R/r)), with R the centreline radius and r the tube radius.

Three caveats come with the formula, and they matter later. It assumes an isolated ring in free space, a smooth surface, and dry air. A real string adds the end fitting, the sheds, and grounded tower steel, all of which push local gradients upward. The screening pass is therefore a floor, never a finished answer.

Now run the numbers with Û = 180 kV and a trial centreline R = 20 cm. Each trial radius also shifts its own onset limit, because the curvature term grows as the tube gets thinner. The table below is the whole first pass.

The screening floor for this 220 kV example sits between 10 and 12 mm of tube. In our experience that number surprises people who expect the calculation to hand over the finished ring. It cannot, because the model has not yet seen the fitting, the tower, or a rain shower. What the first pass has done is kill the 8 mm trial before anyone bends aluminum around it.

Sizing Ring Diameter and Position Around the Fitting

With a tube floor established, the second job is geometrical. The ring must project far enough outward to intercept field lines that would otherwise end on the line-end fitting. Hold it the way you would hold a shield just past the object it protects, not against it. Its centre plane should sit level with the top of that fitting, where the stress peak lives on a composite string.

For the example, the envelope of the dead-end eye and the first shed sets a trial centreline radius near 20 cm. The loop later settles on R = 21.5 cm with a 45 mm tube. The outer diameter then works out to 2 × (21.5 + 2.25) ≈ 475 mm. In this layout the tube centre plane sits at the fitting top face and the rim projects about 150 mm beyond it.

Position also has a negative rule. The ring must not barge into the clearances that the line design already reserves between phase hardware and grounded tower steel. If the layout sketch cannot keep that distance, the ring diameter is not the negotiable item; the bracket geometry is. Hardware context for the string side of this layout appears on the aislador para línea aérea line-up.

What the Altitude Correction Changes

Air density enters the Peek relation through δ, the ratio of local density to its sea-level value. Standard-atmosphere tables put clear-air density at 2,000 m near 1.007 kg/m³, against 1.225 kg/m³ at sea level, so δ ≈ 0.82 at the mountain site. Thinner air ionizes sooner, and the calculation has to pay for that before the geometry ships.

Run the variant with the settled geometry. For the 45 mm tube, the sea-level onset of 36 kV/cm falls to about 30 kV/cm at the mountain site. The ceiling drops from 27.4 to 22.9 kV/cm, about 16 percent. The screening field estimate itself does not grow, but the acceptance band around it shrinks. That is the whole altitude story in one line: same ring, thinner margin.

A design that clears its ceiling with seconds to spare at the coast can fail the same check on a mountain pass. When the site altitude is known, the correction costs one line of arithmetic. When it is unknown, the honest move is to size for the plateau and note the assumption on the drawing.

Closing the Loop With Field Simulation

The simulation loop is where a corona ring design becomes defensible. The electrostatic model now includes the full assembly: both end fittings, the housing sheds, the ring and its yoke, and the nearest grounded steel at true spacing. The model is energized at the crest operating voltage, and the solver returns the peak surface gradient on every metal edge, not just on the ring.

The acceptance test inside the loop is the ceiling from the second section. If any peak sits above it, the loop answers with geometry: a fatter tube, a wider projection past the fitting, or a shifted centre plane, and then re-solves. For the 220 kV example the loop settles at a 45 mm tube on a 21.5 cm centreline. That is the 475 mm ring quoted above, with the plane level with the fitting top. Those values clear the ceiling with room for weather, which is precisely what the margin was reserved for.

Have a String That Needs Its Own Numbers?

A custom-engineered corona ring starts from your voltage class, string drawing, and site altitude, and comes back as a dimensioned proposal with the calculation behind it.

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Laboratory Proof: Extinction and RIV Acceptance

Calculation and simulation end at a predicted field; the laboratory ends at a measurement. Fittings for overhead lines above 45 kV sit under IEC 61284, Overhead lines – Requirements and tests for fittings. Its scope explicitly reaches these products and extends to similar substation fittings. A test certificate against that framework is the usual closing evidence that the calculated geometry survives contact with wet, real air.

Two observations carry most of the weight in that proof. The extinction check asks whether visible corona dies at or below the specified voltage in a darkened hall. The radio-interference measurement asks the same question in a frequency band that television viewers care about. Both are pass-or-fail against the limits the standard and the purchase order call up, and both are run on the assembled fitting, ring included.

When the Numbers Force a Redesign

By the time a crew hears a hiss from the tower top, the failure is already legible in the arithmetic. Every corona ring design failure mode traces back to a skipped or optimistic check, and the worked example above makes each one concrete.

• Tube too thin: the 8 mm trial already failed the ceiling check on paper. Built anyway, the field crowds around it, the air breaks the same way it would without a ring, and the line-end fitting keeps glowing at dusk.
• Ring oversized: diameter buys field relief but charges weight, wind load, and bracket moment. Past the point the calculation justifies, it also eats the clearance to tower steel that the layout must keep.
• Plane misplaced: a ring set below the fitting top shields air instead of hardware. The stress peak simply migrates to the first shed, and the string develops corona one fitting-length away from the shiny new ring.

None of these reach the tower by accident; they reach it through shortcuts. The ceiling check, the envelope check, and the plane check together cost less than one re-energized outage. Hardware on the string side that carries these loads is covered under accesorios de tensión y suspensión.

Handing the Numbers to the Factory

A corona ring design reaches the factory as a drawing package, not as a conversation. The sheet that leaves this walkthrough carries the settled geometry. It lists the 45 mm tube, the 475 mm outer diameter, the centre plane at the fitting top face, and the bracket interface dimensions. Radial dimensions are called out to ±1 mm in this example, because a rim that wanders off-centre shifts the very field the calculation tuned.

The sheet also carries two surface notes that protect the assumptions. The ring is specified in aluminum alloy, formed as a smooth toroid, because the irregularity factor m = 0.95 was an input, not a hope. Weld seams and bracket transitions are called out as dressed and blended, since a sharp bead re-creates the sharp edge the ring exists to remove. A drawing that omits those notes invites a ring that measures right and behaves wrong.

Conclusión

Corona ring design is a chain of small verifications, and the 220 kV walkthrough above kept every link visible. A field limit came first, and the tube trial cleared or failed it. Diameter and plane answered the fitting, altitude taxed the margin, and simulation closed the loop before the laboratory signed the last page. Run the chain the same way on your own voltage class and the hiss never happens. RaxPower manufactures corona rings and matching line hardware to drawing, and treats that chain of verifications as the shortest distance between a specification sheet and a quiet energized line.

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