Calculating a Lightning Rod’s Protection Zone: Methods and an Example

Three methods for determining the protection zone

• the rolling sphere method — universal, applicable to structures of any shape: an imaginary sphere of a given radius is “rolled” over the structure and its air terminations; the protected zone is wherever the sphere cannot touch;
• the protective angle method — a simple approach for buildings of simple shape and limited height: the protection zone is a cone from the air termination at a given angle to the vertical (the angle shrinks as height grows, and the method itself works only up to 20–60 m depending on the class);
• the mesh method — for protecting flat surfaces (for example, large roofs): a mesh of conductors with a set cell size is laid over the surface.

Parameters by lightning protection system class

The LPS class (I–IV) is chosen from a risk assessment (DSTU EN 62305-2); class I is the highest level of protection, IV the lowest. Sphere radius and cell size come from Table 2 of DSTU EN 62305-3:2021 (cl. 5.2.2) and are the largest permitted values. The height limit for the angle method is from Figure 1 of the same standard:

For the protective angle, Table 2 only refers to Figure 1; see the angles and the height limit in “Protection angle method: up to what height it applies” below.

Example: protection zone of a single rod air termination

For a single vertical rod of height h (less than the sphere radius r) on flat ground, the ground-level protection radius under the rolling sphere method is R = √(h × (2r − h)).

Example: a 6 m rod, LPS class III (sphere radius 45 m): R = √(6 × (2×45 − 6)) = √(6 × 84) = √504 ≈ 22.4 m.

Multiple air terminations: a shared protection zone

The zones of two or more rods can overlap. The rolling sphere method shows whether the sphere can touch the object between neighbouring air terminations: if it can, an unprotected area remains between them and another rod is needed. The formulas for two rods of equal height are in “Two rods of equal height: the largest distance between them” below.

What we need to calculate your object

Send the type and dimensions of the object to be protected, whether other structures or equipment are nearby, and the risk assessment result (LPS class), if already done. We will propose a lightning rod design — free-standing or combined with a lighting pole or telecom tower.

Example: a mesh for a flat roof

The cell size in the standard is a maximum, so the number of cells is rounded up. For a flat rectangular roof of 60 × 30 m and LPS class III (15×15 m mesh): 4 × 2 cells (4 along the 60 m side, 2 along the 30 m side), conductor length (2 + 1) × 60 + (4 + 1) × 30 = 330 m. The outer conductors run along the roof edges, so each direction has one more conductor than cells. This counts the mesh alone, without down-conductors or a connection allowance. For all classes:

Class IV: 30 ÷ 20 = 1.5, so 2 cells of 15 m are used. For this roof, class I needs 2.7 times as much conductor as class IV.

Checklist and typical mistakes

• the LPS class is determined by the risk assessment under DSTU EN 62305-2 (see lightning protection norms);
• the number of cells is rounded up, otherwise the cell size would exceed the permitted maximum;
• on a roof sloping more than 1/10, a conductor is also run along the ridge; instead of a mesh, parallel conductors spaced no wider than the cell can be laid;
• a condition for using a mesh (Annex A, cl. A.3): lightning current must always have at least two different metal paths to earth.

Protection radius by class: rods of 8–15 m

The formula R = √(h × (2r − h)) works better as a table. It gives the protection radius at ground level for a single rod; at the height of the object the zone is narrower, and the article Lightning protection of masts and towers shows how to work it out.

Radius in metres at ground level. The formula holds while the rod height does not exceed the sphere radius r of the class.

How tall the rod must be to protect an object at a distance

The inverse problem: an object of height hx stands at a distance x from the rod. By the rolling sphere method it is protected if x ≤ √(h × (2r − h)) − √(hx × (2r − hx)). From this the rod height is h = r − √(r² − (x + √(hx × (2r − hx)))²).

Example: class III (r = 45 m), an object 5 m high at 12 m from the rod. √(5 × 85) ≈ 20.6; 12 + 20.6 = 32.6; h = 45 − √(2,025 − 1,063) ≈ 45 − 31.0 = 14.0 m. The result is rounded up. For an object 5 m high it looks like this:

Rod height in metres. A dash means the sphere radius is too small and a single rod cannot cover the object. For class I under these conditions one rod is not enough even at a distance of 8 m.

Two rods of equal height: the largest distance between them

A sphere resting on the tips of two rods sags between them by p = r − √(r² − (d/2)²) (formula E.2 of informative Annex E), where d is the distance between the rods (no more than 2r). The article Lightning Protection of Masts and Towers has an example with a sag table for two masts.

An object between the rods is protected if it is lower than h − p. From this, our own derivation gives the largest distance for an object of height hx: with p = h − hx, d = 2 × √(p × (2r − p)). For class III (r = 45 m) and rods of 10 m: a 4 m object gives p = 6 m and d = 2 × √(6 × 84) ≈ 44.9 m, and a 6 m object gives d ≈ 37.1 m. This checks the line between the rods only; the lateral limits of the zone need a full calculation.

For three or four rods the formula is the same; for four rods at the corners of a square, d is the square’s diagonal (Figure E.20 of informative Annex E).

Protection angle method: up to what height it applies

Under DSTU EN 62305-3:2021 (cl. 5.2.2, Table 2, Figure 1) the angle method is limited by height. The curves in Figure 1 for classes I–IV end at about 20, 30, 45 and 60 m, like the sphere radii in Table 2; above that the method is not suitable. Note 1 to Figure 1 names only the rolling sphere and mesh methods beyond it. The standard calls a mesh acceptable for flat surfaces, so in our view a free-standing mast is left with the sphere.

The height h is measured from the reference plane of the protected area, that is, the surface being protected: for a free-standing mast, the tip above the ground; for a mast on a roof, the height above the roof if the roof is protected, above the ground if the ground beside the building is (Figure A.2 of normative Annex A, Figure E.12 of informative Annex E).

The standard gives the angle only as a graph (Figure 1), with no table. Up to a height of 2 m it does not change (roughly 70° for class I to 79° for class IV), then it falls to about 23° at the height limit in all four classes. This is a rough reading from the graph: for a project, take the exact angle from Figure 1 of the official text. By our own calculation (R = h × tan α, where R is the ground-level radius), at the class II limit of 30 m a mast with an angle of 23° has a protection radius of about 13 m, while the rolling sphere method gives 30 m.