26 th
-30 th
November 2007 – Foz do Iguaçu, Brazil
W.A. Chisholm A. Phillips
Kinectrics / University of Quebec at Chicoutimi
W.A.Chisholm@ieee.org
EPRI aphillip@epri.com
G1-1050 555 Boulevard de l’Université, Chicoutimi, Québec G7H 2B1 CANADA
Abstract - The low-frequency resistance of the wire-frame approximation to a geometric solid of any shape can be expressed as the sum of two terms: a geometric resistance, which varies inversely with the geometric radius of the solid, and a contact resistance, which varies inversely with the total length of wire in the frame. Analysis and comparisons with analytical and numerical results show that a fill factor based on the logarithm of the ratio of figure area to wire area provides a good estimate of contact resistance. buried radial wires that can benefit from the introduction of a multiplicative constant, varying slowly with the ratio of overall area to wire area to modify the Laurent contact resistance term.
Analytical expressions [4-6] can be used to calculate the ground electrode resistance of relatively simple shapes or regular forms. However, these expressions may only be
The use of two separate terms in the calculation of the resistance of substation ground electrodes has been common for many years. These two terms are now labelled “Geometric” and “Contact” resistance. A wellknown example in calculations for substation grounding
[1] uses a solid disc with radius giving the electrode area
A , along with the total length of buried wire, uniform soil resistivity,
ρ
in Eqn. (1):
R
=
R
1 INTRODUCTION
Geometric
+
R
Contact
=
4
ρ
A
π
+
ρ
L
L , and the accurate in a limited range of electrode dimensions.
Ground electrode resistance can also be calculated using numerical methods such as the method of moments [7,8] in computer programs. Both analytical [4-6] and numerical [7,8] methods take a considerable effort to set up the equations or input the data respectively.
Commercial programs for accurate substation grounding computations are expensive and take considerable experience to learn. Neither approach can be inverted or manipulated in closed form to work out the length of buried wire or number of rods needed to achieve a target resistance or an optimum electrode cost. Neither method
(1) gives much physical insight into how different choices
(wire size, wire length) affect results.
This paper first reviews calculation of the geometric resistance term in Eqn.(1) and makes the definitions of electrode size and area more precise. Then, the contact
Improvements in the calculation of geometric resistance have been proposed [2] with the use of maximum extent in any direction ( s
1
) and area A . The geometric resistance expression was later updated [3] to use geometric radius resistance in Eqn. 1, being the difference between geometric resistance and total resistance of a wire frame
( s
3
) and area A . An upper bound of the Laurent contact resistance term in Eqn.(1) as ( R
Contact
≤ ρ
/ L ) was used in the updated model [3] for wire frame approximations to solids, such as a rectangular grid and a metal box of the same shape.
For many practical transmission tower ground electrodes, the sum of the geometric resistance of the solid electrode, and the upper bound of contact resistance ( R
Contact
=
ρ
/ L) gives a good estimate of total resistance. The upper bound tends to over-estimate the resistance, but this term is approximation, is normalized using the Laurent form to give K=L R
Contact
/
ρ
. For many different electrode shapes and geometries, K turns out to have the same relation to the natural logarithm of the fill ratio of wire area A wire
to electrode area A .
The new method is:
•
Easy to describe and fast to implement
•
Sufficiently accurate for many different geometries and for wide variations of the shape of each electrode
•
Invertible for most geometries
•
More accurate than some analytical expressions usually small compared to geometric resistance.
However, there are some important electrodes such as when compared to reference numerical method solutions.

2 GEOMETRIC RESISTANCE
The concept of a simple expression that would accurately describe the resistance of a solid electrode, with a wide variety of shapes, [2] was originated to model the changing shape of the volume of ionization around ground electrodes under lightning surge conditions [9].
The model was concieved with the following features:
•
Accuracy of about 10%, comensurate with the
• soil resistivity estimate
Gives R
Geometric
=
ρ
/2
π r for hemisphere exactly, where r is the radius of the hemisphere
•
Gives R
Geometrc
that is asymptotic to one of the derived expressions for resistance of a thin vertical rod [4-6]
•
Expresses ratio of electrode size s to surface area
A as a slowly-varying shape factor as in Eqn.(2):
R
Geometric
=
2
ρ
π s ln
⎛ ⋅
⎜⎜ k
A s
2
⎞
⎟⎟ (2)
Values of the semi-empirical constant k were obtained
[2,3] by selecting specific measures of electrode size s
(radius [2] or geometric radius [3]) and then inverting
Eqn. 2 using the resistance of a hemisphere.
2.1 Measures of Electrode Size
In [2], the electrode size was measured as s , “the distance from the centre of the electrode to its outermost point”.
This definition was clear for some electrodes (such as a vertical rod, s=rod length or a hemisphere, s=radius ) but ambiguous for other shapes. Here are some possible interpretations of electrode extent: s
1
=
MAX ( R
X
, R
Y
, R
Z
) s
2
=
MAX ( R
2
X
+
R
Y
2
, R
2
X
+
R
Z
2
, R
Y
2 +
R
Z
2
)
(3) s
3
=
R
2
X
+
R
Y
2 +
R
Z
2
Reference [3] used the s
3
dimension and derived its appropriate value of k . In practice, s
3 was used for rectangular electrodes as a logical interpretation of the original definition of s in [2].
The surface area A was defined in both [2,3] as the total
(bottom and side) area of the hole that would be excavated to expose the entire electrode.
2.2 Geometric Shape Factor
Using the conditions above, the constant k for each definition of maximum electrode extent s
1
, s
2
and s
3 varies from 17 to 11.8 as shown in Table 1. The Sunde
[5] expressions for resistance of a thin vertical rod and thin disc are considered as accurate for this case:
R
Rod
=
ρ
2
π l ln
⎛
⎝
2 l r
⎞
⎠
; R
Disc
=
ρ
4 r
(4)
These are also shown in Table 1 for comparison. While all three measures of maximum extent give reasonable approximations to the rod and disc resistance, s
2
and s
3 work better than s
1
.
Table 1: Geometric resistance expressions for Eqn.(2) for three definitions of maximum extent in Eqn.(3) s
1 s
2 s
3
Geometric
Resistance
2
π
ρ s
1 ln
⎜
⎜
⎝
⎛
17 s
1
2
A
⎟
⎟
⎠
⎞
2
π
ρ s
2 ln
⎛
⎜⎜
13 s
2
2
A
⎞
⎟⎟
2
π
ρ s
3 ln
11 .
8 s
3
2
A
Equivalent:
Vertical
Thin Rod s
1
= l
R
=
ρ
2
π l ln
⎛
⎜
⎝
2 .
7 l a
⎞
⎠ s
2
R
=
= l
ρ
2
π
L ln
2 .
1 l s
3
= l
R
=
ρ
2
π l ln
1 .
9 l
radius a a a
length l
Equivalent:
Hemisphere
radius r s
1
R
=
= r
ρ
2
π r s
2
R
=
=
ρ
2
⋅
2
π r r s
3
R
=
=
ρ
3
⋅ r
2
π r
Equivalent:
Thin Disc s
1
R
=
=
radius r
* 17 = 2
π e ; 13 = r
0 .
269
π e
ρ r s
2
R
=
= 0
2
⋅
.
237
√
2
; 11.8 = (2
π e r
ρ s
3
R
=
=
2
⋅ r
0 .
227 r
√
3
)/3; e = 2.718
ρ r
2.3 Geometric Resistance of Smooth Electrodes
Sunde [5] gives a reference expression for the resistance of an ellipsoid of revolution of semi-major axis l and semi-minor axis a in Eqn.(5):
α
R
=
=
α
ρ
4
π o
α l
1
[
α
+
−
+
1
1 ln
( )
1
−
4
α
2 o
2
]
α o
=
2 l a
2 −
2
(5)
Fig. 1 – Ratio of geometric resistance to reference value for [5] using three definitions of maximum extent s

Fig. 1 shows the ratios of the geometric resistance expressions using the possible definitions of maximum extent ( s
1 s
2 s
3
) in Eqn.(3) and Table 1 to the reference value from Eqn.(5) for a wide range of aspect ratio.
Neither the disc or rod expression in Eqn.(4) provides a useful estimate of the resistance of a solid electrode that is roughly hemispherical. The estimate using the maximum extent in a single direction s
1
is not much better in this respect, with errors of up to 23%. However, the 2-D extent s
2 and 3-D extent s
3
both give geometric resistance values that are within
±
8% of the reference Eqn.(5).
2.4 Geometric Resistance of Rectangular Electrodes
For rectangular electrodes of length l , width w and depth
R
Z
the dimensions R
X and R
Y in the horizontal plane are semi-axes of an ellipse with perimeter P=2(l+w) , given in
Eqn.(6). The area A in this case is lw+PR
Z
.
R
X
R
Y
=
=
1
2
(
6 D l
2 + w
2
−
8 R
X
)
+
(
6 D
D
−
8 R
X
=
2 ( l
)
2
12
−
+ w )
24
(
D
π
2 +
6 R
2
X
−
6 DR
X
)
(6)
2.5 Measures of Electrode Area for Wire Frames
The removal of interior volume from solid metal objects
(boxes, cylinders) was shown by Chow and Srivastava
[10] to make only minor changes to the self capacitance of the objects. Up to 90% of the metal skin can be removed before a 10% change in capacitance is computed. Since there is a dual relation between the capacitance of an object and its resistance, the same principle is true for wire frame approximations to solid electrodes. This means that a wire box and a solid cube of metal with the same corner dimensions have the same surface area, the same extent and thus the same geometric resistance.
When two buried wires cross at different directions, there is ambiguity in the definition of solid electrode area A .
Fig. 3 shows that the sum of two trench areas (two sides, bottom, two ends of each) could be used. However, the total area (four sides plus base) of the quadrilateral formed by the ends of the wires, shown in Fig. 4, is correct for evaluation of the geometric resistance
A series of reference numerical calcluations using a moment-method approach [7] were carried out to compute the accurate resistance of metal boxes, with a
50:1 range in l/w and a 10:1 range in l/R
Z
in soil of resistivity
ρ
=100
Ω m. These computational results are seen in Fig.2 to be in close agreement with the geometric resistance estimates using either the s
2 and s
3
estimates of electrode size. The s
1
dimension has greater scale error
(8.12%) and a smaller Pearson coefficeint R
2
of 0.979 in its linear regression.
Fig. 3 – Possible interpretation of the surface area A for two buried wires in calculation of geometric and contact resistance
Fig. 2 – Comparison of geometric resistance, Eqn.(2) with numerical calculations [7] for metal boxes of various shapes
Three definitions of maximum electrode extent s from Eqn.(3)
Fig. 4 – Preferred surface area A of two buried wires in calculation of geometric resistance using Eqn.(2) and Eqn.(8)
The reason for this choice of Fig. 4 becomes clear when the asymptotic behaviour of a large number of radial wires is considered. The surface area of the trenches in
Fig. 3 becomes infinite, while the area in Fig. 4 approaches that of a solid cylinder as desired.

3 CONTACT RESISTANCE: BURIED WIRE
The difference between the true resistance of the wire frame from a reference expression or calculation and the geometric resistance is the contact resistance. We normalize this difference using the Laurent [1] approach with, R
Contact
=
ρ
/L, for grids for total wire length L :
K
=
L
( R reference
−
R
Geometric
)
(7)
It is instructive to plot the value of K in Eqn.(7) for some typical wire frames, and to then establish an expression for K . In [12], individual expressions were fitted to K for each geometry, such as ring, wire or multiple radial counterpoise. However, it is more satisfactory to obtain a single expression, independent of geometry, as follows.
The simplest demonstration of this approach uses the buried wire as a wire-frame approximation to a metal plate, set vertically in the soil. Fig. 5 shows that K varies slowly from zero for fat wires near the surface to slightly over unity for thin wires buried at considerable depth.
Where A
Wire
is 2
π r(L + r) , counting wire ends and length and A =
π rL+2(L+r)R
Z
, the same area of the solid approximation to the electrode trench that is used in
Eqn. 2 to compute the geometric resistance.
4 CONTACT RESISTANCE: RODS IN A CIRCLE
The Eqn.(8), derived as a good fit to the variation of K with fill ratio A
Wire
/A for a single horizontal wire, turns out to be an equally accurate and robust estimate of contact resistance for many other electrode geometries.
Numerical verifications of derived equations for one to sixteen vertical rods in a row or circle and two to twelve radial wires were carried out at the same time using [7].
As a demonstration, the Sunde [5] expression in Eqn.(9) for resistance was used as a reference check. For n rods of length l and radius a in a circle of diameter D , [5] gives series sum and approximate resistance expressions as:
R n
R n
=
≈
1 n
1 n
ρ
2
π l ⎣
⎢
⎡ ln
4 l a
ρ
2
π l
⎡
⎢⎣ ln
4 l a
−
1
+ l
D m
=
∑
−
1
+
2 l ln d n
−
1 m
=
1 sin
π
1 m
2 n
π
⎤
⎥⎦ n ⎦
⎥
⎤
(9)
In Eqn.(9), d is the separation between rods. Fig. 6 shows that some resistance values from Eqn.(9) are in relatively poor agreement with values computed numerically [7], in cases where the spacing and rod length become equal. In contrast, the resistance values obtained from the sum of geometric and contact resistance (Eqn.(2) and Eqn.(8)) using the s
3
dimension remain in close agreement with the numerical reference values for all cases.
Fig. 5 – Variation of Contact Resistance Multiplier K in Eqn.(7 ) with logarithm of fill ratio for buried horizontal wire
Fig. 6 – Comparison: Sunde expressions and wire frame method
(Geometric + Contact Resistance) for n vertical rods in a ring.
Reference values: computation using moment method [7]
There is little influence in the choice between s
2
and s
3
for this estimate of K . Therefore a suitable expression for both cases is:
K
=
MAX
1
2
π ln
⎛
⎜⎜
2
⋅
A
A
Wire
⎞
⎟⎟ , 0 R
Contact
=
K
L
ρ
The resistance estimate using Eqns.(2) and (8) is robust, compared to the analytical expressions [5], meaning that
(8) it retains good accuracy in degenerate cases such as when rods or radial wires are close together, widely separated, have a large radius or are buried at shallow depth.

5 CONTACT RESISTANCE: IONIZED WIRE
Contact resistance of wire frame electrodes is influenced by changes in wire radius, including those associated with ionization under lightning surge conditions. In contrast, geometric resistance of the wire-frame approximations to typical grounding electrodes is insensitive to changes in wire radius. This means that the separation of resistance into two terms also gives engineers a tool to evaluate the effects of ionization on complicated electrodes consisting of large foundations and buried wires in parallel.
Recently, He et al [11] presented their analysis of the effects of ionization on the resistance of a buried wire.
Under lightning surge conditions, the local electric field gradients can reach levels that establish a limiting gradient of somewhere between 300 kV/m and 1000 kV/m. The equivalent radius of the ionized zones varies along the wire length, as described in Fig. 7.
Fig. 7 - Modelling of equivalent radii of ionized zones around buried horizontal counterpoise wire [11]
The effects of ionization on the geometric and contact resistance terms of an electrode with different radii at each segment were evaluated and compared to the total value from numerical methods [7]. For a 30-m buried wire with no ionization, computed resistance of 5.8
Ω reduced to 4.3
Ω
when 10 m of the wire was modelled as an exponentially decreasing radius with a maximum of 1 m at the centre of the wire. Fig. 8 shows that there was only a 3% reduction in the geometric resistance term with the increased radius, while the contact resistance term decreased by about 78%.
Fig. 8 – Relative decrease in contact and geometric resistance for three sets of equivalent ionization radii
This implies that the separation of wire frame resistance into the geometric and contact resistance terms, as outlined above, also provides an efficient means to establish the upper bound of the effect of ionization on the resistance of the electrode. Contact resistance will tend to become vanishingly small, while there will be little change to the geometric resistance.
6 CONCLUSION
The resistance of any solid object can be well approximated using a measure of the electrode size ( s ) and its surface area ( A ) using Eqn. (2). It does not make much difference whether a two-dimensional distance s
2
or a three-dimensional distance s
3
is used, as long as the appropriate adjustment is made in the expression to give the resistance of a hemisphere exactly.
The difference between the resistance of a wire frame approximation to a solid electrode, and the solid object, can be expressed efficiently as a contact resistance that varies inversely as the total length of the wire frame L , adjusted with a “fill factor” given by the logarithm of the ratio of overall area A to wire area A
Wire
.
Combining Eqn.(2) and Eqn. (8), the resistance of a wire frame electrode is:
R
=
2
ρ
π
⎛
⎜
1 s
3 ln
⎛
⎜⎜
11 .
8 s
3
A
2
⎞
⎟⎟
+
1
L ln
⎛
⎜⎜
A
2 A
Wire
⎞
⎟⎟
⎞
⎟
(9)
Eqn.(9) is more accurate than Sunde’s expressions for multiple rods in a circle, compared to reference computations using numerical moment methods [7].
The separation of geometric and contact resistance terms allows for efficient analysis of soil ionization under lightning surge conditions. Ionizatoin will tend to reduce the contact resistance but not the geometric resistance for typical large electrode systems.
7 REFERENCES
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Boca Raton, FL, RC Press, 2000.
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Systems,
Earth Conduction Effects in Transmission
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EPRI
1,2)”,Final Report EL-2699 for EPRI Research Project
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