The Development of a 3-D Rolling Sphere

Algorithm for Lightning Protection

Neil McDonagh

ESBI

neil.mcdonagh@esbi.ie

Danijela Klopotan

ESBI

danijela.klopotan@esbi.ie

Abstract-When designing Lightning Protection Systems (LPS), the effectiveness of an air termination structure(s) may be assessed by using the rolling sphere method. This method is outlined in standards BS6651:1999 [1], IEC 62305 [2], and IEEE 998-1996 [3]. A number of 2D methodologies are presented in these standards illustrating how to apply the rolling sphere method. This paper proposes a new 3D approach to the application of the rolling sphere method.

This new approach analyses the designated air termination

structures and forms stationary shapes around the air

terminations. This approach focuses on where a rolling sphere

can not go rather than where it can go. For example, a concave

cone, as shown in Figure 1, will describe the protected area

provided by a single lightning mast. Different shapes are

required depending on the nature of the air termination system.

All shapes will be a combination of sections of concave cones,

spheres and cylinders.

Index Terms-- air termination system, lightning protection, rolling sphere method.

I. INTRODUCTION

Lightning protection systems (LPS) can be broadly

categorised as having three components; air termination

structure(s), down conductors and earth terminations. The

effectiveness of the air termination structure(s) is assessed by

using the angle of protection method or in the case of taller

structures the rolling sphere method. A good description of

the application of the rolling sphere method is: “Use of the

rolling sphere method involves rolling an imaginary sphere of

radius S over the surface of the substation. The sphere rolls

up and over (and is supported by) lightning masts, shield

wires, substation fences, and other grounded metallic objects

that can provide lightning shielding. A piece of equipment is

said to be protected from a direct stoke if it remains below the

curved surface of the sphere by virtue of the sphere being

elevated by shield wires and other devices. Equipment that

touches the sphere or penetrates its surface is not

protected”[3]

. The application of the rolling sphere method is

outlined in BS6651:1999 [1]

and IEC 62305 [2]

, IEEE 998-

1996 [3]

, and a number of 2D methodologies are presented in

order to apply the rolling sphere method.

TABLE I PROTECTION ZONES USING 3D STATIONARY SHAPES

Protective

Device

Angle of Protection

Method

Rolling Sphere Stationary

Shapes

Mast Cone Concave Cone

Flat Wall Prism Cylinder Section

Cylinder Conical Frustum Concave Conical Frustum

Fig. 1. 3D-Stationary Shape for a Single Lightning Mast

This paper proposes a new 3D approach to the application of

the rolling sphere method. The new method relies on

stationary shapes rather than rolling spheres. In this regard it

is similar to the angle of protection method. However the

shapes used are more complex. Table I defines the shape of

the zone of protection for different types of air termination

structures. All possible zones of protection, even arising from

a combination of different air termination structures in close

proximity to each other, may be defined by the combination

of sections of concave cones, cylinders and spheres. For

example the protected area of a single lightning mast may be

described by a concave cone constructed around a lightning

mast, as shown in Figure 1.

II. EXISTING APPLICATION OF THE ROLLING SPHERE METHOD

A. 2D Methods – Section View

Taking a plan drawing of the installation to be protected, a

number of paths may be selected, over which the rolling of a

sphere of appropriate radius is considered, which may be seen

in Figures 2 and 3.

Fig. 2. Rolling Sphere Paths over Installation

Lig

htn

ing

Ma

sts

Fig. 3. Rolling Sphere Method applied in a 2D Plane

One drawback of using this method is encountered when

considering a rolling sphere path that does not pass directly

over a lightning mast or infrastructure to be protected. In this

instance it is necessary to consider the effective height of the

lightning mast or the protected equipment. In order to

calculate the effective height of a lightning mast the following

methodology may be used. A sphere considered in a 2D plane

is a circle described by Equation 1.

( ) ( ) 222rkyhx =−+− (1)

Where:

(x, y) is a point on the circumference of the circle

(h, k) is the centre of the circle and

r is the radius of the circle

An illustration of this 2D view of the rolling sphere is shown

in Figure 4. The sphere is considered to be rolling into the

page, where the centre point of the sphere will not pass

directly over the lightning mast. If a rolling sphere is

impinged upon by a foreign object its path must change. In

our analysis the path of the sphere on the X axis does not

change. Therefore the sphere must move along the Y axis to

avoid the obstacle as shown in Figure 4.

When the sphere is impinged upon it will rise to a height of

(k-r) above the ground. Therefore the effect of the object at

point (x, y) may be replaced by a different object in the path

of the sphere with effective height (k-r). This principle is used

to calculate the effective height of the lightning masts within

the substation compound for different paths of the rolling

sphere.

The equation of a circle must be rearranged to solve for k, see

(2)-(6). Solving the quadratic equation (4) gives two solutions

one where k>y and one where k<y. Therefore the solution

depends on the height of the lightning mast in relation to the

radius of the rolling sphere. For the purposes of this analysis

it is considered that the centre point of the sphere is higher

than the height of the lightning mast k>y (see Equations 6a

and 6b). It can be seen that k-r is the effective height of the

lightning mast seen by the sphere if the lightning mast is

transferred directly into the path of the sphere. The effective

height of two lightning masts (such as in Figure 2), is

calculated and presented in Table II. A close-up view of a

substation with the rolling sphere method applied in a 2D

plane is shown in Figure 5.

Fig. 4. 2D view of Sphere with Lightning Mast

( ) 0hxryyk2k2222

=−+−+− (2)

( )222 hxryZ −+−= (3)

0Zyk2k 2=+− (4)

2

Z4y4y2k

2−±

= (5)

2

Z4y4y2k

2−+

= (6a)

22 )( hxryk −−+= (6b)

TABLE II EFFECTIVE MAST HEIGHT (METERS)

Mast 1 Mast 2

Path

Ro

llin

g S

ph

ere

rad

ius

–(r

)

Mas

t H

eig

ht

–

(y)

Dis

tan

ce f

rom

pat

h –

(x

-h)

Eff

ecti

ve

hei

gh

t

(k-r

)

Mas

t H

eig

ht

–

(y)

Dis

tan

ce f

rom

pat

h –

(x

-h)

Eff

ecti

ve

hei

gh

t

(k-r

) AA 60 20 0 20.00 15 0 15.00

BB 60 20 20 16.57 15 15 13.09

CC 60 20 25 14.54 15 0 15.00

DD 60 20 7 19.59 15 10 14.16

Fig. 5. Rolling Sphere Method applied in a 2D Plane

(h , k)

(x , y) Lightning

Mast

k-r X

Y

While this method is powerful, it can be time consuming to

apply, and leads to the creation of many separate drawings.

The method is only as accurate as the number of paths chosen

for analysis. Some skill may be required to choose the paths

that will lead to the identification of unprotected equipment.

B. 2D Methods – Plan View

Considering a substation or other installation that needs to be

protected, taking account of the highest piece of plant and the

height of lightning masts, it is possible to assign a protective

radius to each lightning mast. Inside the circle of this radius

all items of plant below the specified height are considered to

be protected. If there is a large variation of the height of items

of plant then a number of different radii may be considered.

An illustration of the determination of the protective radius is

shown in Figure 6.

22 EErrP −+= (7)

22 22 EEryryP −−−= (8)

Where: P is the protective radius

r is the radius of the rolling sphere

E is the maximum height of the equipment

y is the height of the lightning mast

The equations to describe the protective radius can vary

depending on the height to the lightning mast compared to the

radius of the rolling sphere. Where ry ≥ , Equation (7)

should be used, and where ry ≤ , Equation (8) should be

used.

An illustration of the application of this method is shown in

Figure 7. While this method is extremely useful, drawbacks

include an over-simplification of the protection area between

different lightning masts, although some methods for

addressing this issue are outlined in [3]. It may also be

cumbersome to design an optimal air termination system

when the objects to be protected are not of a uniform height.

C. 3D Method – Collection Surface

This Method has been proposed by Q Xie et al, in a paper

titled “Rolling Sphere Method using 3D Graphics Approach”

[4]. An illustration of the application of this method is shown

in Figure 8.

This approach does provide an extremely powerful 3-D

method for defining the protection area associated with a

rolling sphere. However, it may require the generation of

many different surfaces. It may be visually difficult to

determine what items of plant are actually protected and

which items are not. The application of this method may

require complex programming and 3D graphics software,

which may not be accessible to all lightning protection

professionals.

Fig. 6. Determination of protective radius

Fig. 7. Rolling sphere method using 2D Plan view

Fig. 8. A Simple Substation with Collection Surface [4]

(h,k) (x,y)

Lightning

Mast

X

Y

P

Equipment to

be protected

E

III. PROPOSED METHOD - 3D STATIONARY SHAPES

A. Single Lightning Mast

This method may be summarised as taking the 2D-Plan view

method discussed in Section 2, and calculating the zone of

protection for items at a variety of heights. Each circle will

correspond to a zone of protection for equipment at a certain

height. Taking each circle, and moving it to its corresponding

height, creates the concave cone shown in Figure 1 and

Figure 9. Another way to visualize this concave cone is by the

rotation of an arc of a circle around the vertical axis (lightning

mast). The arc that is chosen is the arc that touches the ground

and the lightning mast. The arc equation is shown in (9).

)a-2ar-r ,usina ,ucosa()z,y,x(f 2= (9)

where: π20 ≤≤ u and ra ≤≤0 ,

r is the radius of the rolling sphere

a is the horizontal distance from the lightning mast

The formula shown in (9) may be used wherever there is a

lightning mast that is taller than the radius of the rolling

sphere to be used. Where the lightning mast is smaller than

the radius of the rolling sphere (9) is not valid. A new

equation must be developed to describe this concave cone.

This equation is described in (10) and (11).

A simple “IF” loop may be used to distinguish which

equation should be used when considering a single lightning

mast. An illustration of the application of this method to an

installation with items of plant of various heights is shown in

Figure 9, where the unprotected items are shown in red.

) Z,usina ,ucosa()z,y,x(f = (10)

( ) ( )2222 arh2hhrh22ar-rZ −−+−+= (11)

Where: 22

20 hrha −≤≤ and π20 ≤≤ u

B. Multiple Lightning Masts

Where two lightning masts are in close proximity to each

other, their zones of protection may impinge on each other,

and provide a greater degree of protection to equipment

between the two masts. This will only happen if the distance

between the two masts is less than the diameter of the rolling

sphere. In this case there are only two possible positions of

the sphere, where the sphere is touching both lightning masts

Fig. 9. Rolling Sphere Method for a Single Lightning Mast

and sitting on the ground. The hollow space left by these

spheres coupled with the concave cones constitutes the zone

of protection of the system.

To find the zone of protection, the arc between the two masts

and common to the two spheres must be calculated. The

position of this arc is calculated in the following way: There

are only two possible positions where a sphere can touch the

ground and both lightning masts. The two centres of these

spheres are calculated using the equation of a sphere (12):

( ) ( ) ( ) 2222rkzqypx =−+−+− (12)

Where (p,q,k) is the centre of the sphere.

There are two known coordinates for (x,y,z) and it is known

that the height of the centre of the sphere is the same as the

radius of the sphere. This enables the calculation of two

possible centres of the sphere, and both p and q must be

solved for in (13) and (14). Using the centres of the spheres it

is possible to construct an additional zone of protection

between the two masts as shown in Figure 10. It may also be

shown that shapes can be developed for more complex air

termination systems involving buildings and more than two

lightning masts.

A2

AC4BBq

2−±−

= (13)

qMLp −= (14)

The variables used in (13) and (14) are defined in the

Equations (15) – (19).

( )

( )21

21

2

2

2

2

2

2

2

1

2

1

2

1

xx2

xxr2zyxzyxL

−

−−−−−++= (15)

( )( )

21

21

xx

yyM

−

−= (16)

2M1A += (17)

11y2ML2Mx2B −−= (18)

=C ( )rz2Lx2zyxqpL11

2

1

2

1

2

1

222−−+++++ (19)

Where ( )111 ,, zyx and ( )222 ,, zyx are the two points

where the sphere touch the lightning masts.

Fig. 10. Rolling Sphere Method for Two Lightning Masts

IV. 3D STATIONARY SHAPES IN CAD PACKAGES

Detailed equations have been presented as to how the areas of

protection can be defined mathematically for use in a

software package such as MATLAB TM

. However the areas of

protection may be defined just as easily using a 3D CAD

package. Figures 11-13 illustrate the application of the

method using a 3D CAD package.

Fig. 11. Rolling Sphere Method applied by ESBI

Fig. 12. Rolling Sphere Method applied by ESBI

Fig. 13. Rolling Sphere Method applied by ESBI

Here an example is given of a single lightning mast, where

the unprotected item of plant is shown in red. The surface of

the zone of protection is defined by the rotation of an arc of

the rolling sphere around the lightning mast. It must be noted

that structures which are not part of the air termination system

have no impact on the shape of the zone of protection, and

unprotected items are identified as those which are outside or

protrude the zone of protection. This method of presentation

can also be coupled with the superposition of local mapping,

see Figure 12. This makes this method extremely easy to

apply, and does not require complex mathematical equations.

V. CONCLUSION

It has been shown that the rolling sphere method can be

applied more effectively by a method which involves placing

certain shapes over objects that form the lightning protection

system (LPS) air termination structure. The method proposed

in this paper defines the surface of the protective area, as

opposed to the consideration of rolling sphere in different

positions. Applying this methodology it is possible to define

the surface of the protected area of an entire LPS, which

includes a number of lightning masts or other protective

devices.

The areas of protection of a lightning protection system can

be defined mathematically in a software package such as

MATLABTM, or graphically in a 3D CAD package. The

benefits of the new methodology over existing 2D and 3D

methods are:

� Better definition of unprotected equipment (2D and 3D)

� Reduction in the number of drawings needed (2D)

� Reduction in computational power needed (3D)

When implementing this solution mathematically, the

definition of the code needed for execution of this method

may be time consuming to construct, but for designers and

engineers who use the rolling sphere method on a regular

basis it may be worthwhile. However, the CAD method is

extremely easy to apply for those who are users of 3D CAD

packages. When applied in a CAD package, the presentation

of the area of protection can be combined with local mapping

to give a clearer view of site orientation, and what items are

not protected.

ACKNOWLEDGEMENTS

The authors would like to thank Ken Atkinson for his

assistance in the production of this paper, and all at ESBI for

their encouragement in the production of this paper.

REFERENCES

[1] BS651:1999 Code of Practice for protection of structures against

lightning [2] International Standard IEC 62305 – 2006 1-4 “Protection Against

Lightning” [3] IEEE std 998-1996 “ IEEE Guide for Direct Lightning Stroke Shielding

of Substations” [4] Q. Xie et al Rolling sphere method using 3D graphics approach”,

Power and Energy Engineering Conference, 2009. APPEEC 2009. Asia-Pacific