For overhead lines operating at high voltages (33 kV and above) use of number of discs connected in series, through metal links, is made. The whole unit formed by connecting a number of discs in series is known as string of insulators. The line conductor is secured to the bottom disc of the string and the top disc is connected to the cross-arm of the pole or tower, as illustrated in Fig. 9.15.
The number of discs connected in series in an insulator string depends upon the line operating voltage (higher the line operating voltage, the larger is the number of discs required for the insulator string, as given below in tabular form).
The number of discs indicated in above Table 9.1 is actually the usual number used. However, in the case of transmission lines operating at 66 kV or more, one disc less than the number indicated in Table 9.1 is used on about eight suspension structures near the substation. This is accomplished so that in the event of a lightning surge appearing on the line, the insulator string will flash-over and prevent the surge from travelling to the substation thus safe-guarding the equipment there.
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It is found that the voltage impressed on a string of suspension insulators (the voltage applied between the line conductor and earth) does not distribute itself uniformly across the individual discs.
The line unit (unit nearest the line conductor) has the maximum value across it, the figure progressively decreasing as the unit nearest the cross-arm is approached. The inequality of voltage distribution between individual units is all the more pronounced with a larger number of insulator units. This fact may be explained with the help of equivalent circuit of an insulator string (Fig. 9.16).
Each string insulator unit behaves like a capacitor having a dielectric medium between the two metallic parts (viz. pin and cap). The capacitance due to two metal fittings on either side of an insulator is known as mutual capacitance. Further there is also a capacitance between metal fitting of each unit and the earthed pole or tower. The capacitance so formed is known as shunt capacitance.
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If a string of similar suspension insulators could be situated so far from neighbouring metal work that the capacitance between this metal work and the metal fitting of the insulators (i.e. shunt capacitance) would be negligibly small in comparison with the capacitance of each unit (i.e. mutual capacitance), then the charging current would have been the same through all of the discs, the discs being connected in series, and consequently the voltage across individual units would have been the same i.e. applied voltage V divided by the number of units in the string.
However, in practice this condition is not fulfilled because of nearness of the tower, the cross-arm, and the line. These shunt capacitances, sometimes called the stray capacitances; have an important effect on the voltage distribution between the units.
Due to shunt capacitance, charging current is not the same through all the discs of the string (Fig. 9.16.). So voltage across individual units, being directly proportional to the current flowing through them, will be different. This unequal potential distribution is undesirable and is usually expressed in terms of string efficiency.
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The ratio of voltage across the whole string and the product of the number of units and voltage across the unit nearest to the line conductor is known as the string efficiency i.e.
where n is the number of units in the string.
String efficiency may also be defined as:
The voltage distribution across different units of an insulator string and string efficiency can be mathematically determined with the help of an equivalent circuit of the insulator string (Fig. 9.16) as below. Fig 9.16 shows the equivalent circuit of a string of suspension insulators containing 4 units.
Let the mutual capacitance between the links be C and shunt capacitance between links and earth be C1, voltage across the first unit (nearest the cross-arm) be V1, voltage across the second unit be V2, voltage across the third unit be V3, voltage across the fourth unit (nearest the line conductor) be V4 and voltage between conductor and earth be V volts.
Let C1/C = K or C1 = KC
Applying Kirchhoff’s first law to node A, we get,
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I2 = I1 + i1
or ω C V2 = ω C V1 + ω C1 V1
or ω C V2 = ω C V1 + ω K C V1
or V2 = V1 (1 + K) … (9.2)
Applying Kirchhoffs first law to node B we get,
I3 = I2 + i2
or ω C V3 = ω C V2 + ω C1 (V1 + V2)
∵ Voltage across the second shunt capacitance C1 from the top = V1 + V2
or ω C V3 = ω C V2 + ω K C (V1 + V2)
or V3 = V2 + K (V1 + V2) = K + V2 (1 + K)
or V3 = KV1 + V1 (1 + K) (1 + K)
∵ From Eq. (9.2) V2 = V1 (1 + K)
or V3 = V1 (1 + 3K + K2) …(9.3)
Applying Kirchhoff’s first law to node C, we get,
I4 = I3 + i3
or ω C V4 = ω C V3 + ω C1 (V1 + V2 + V3)
∵ Voltage across the third shunt capacitance C1 from the top = V1 + V2 + V3
or ω C V4 = ω CV1 (1 + 3K + K2) + ω K C [V1 + V1 (1 + K) + V1 (1 + 3K + K2)]
or V4 = V1 (1 + 6K + 5K2 + K3) …(9.4)
Finally voltage between line conductor and earth,
V = V1 + V2 +V3 + V4
= V1 + V1 (1 + K) + V1 (1 + 3 K + K2) + V1 (1 + 6 K + 5 K2 + K3)
= V1 (4 + 10 K + 6 K2 + K3) … (9.5)
The greatest voltage will be obviously V4 which is given as:
Since n = 4, and flash-over voltage of one unit
= Greatest voltage across any unit i.e. V4
Similarly derivation can be had for a string of insulators consisting of any number of units.
When the number of insulators in the string is large it becomes laborious to work out the voltage distribution across each unit, for such cases standard formula may be used.
In general case if there are n units in the string, V is the maximum voltage across the string, V1,V2, V3…….. Vn denote the voltages across the insulator units starting from top, C is the capacitance between the links and KC be the shunt capacitance between the links and earth, the voltage distribution across the mth unit (counted from top) is given as
and potential adjacent to the line conductor
Graphical plot of how voltages are distributed across the units of an insulator string is shown in Fig. 9.17.
The following points may be noted:
1. The unit nearest to the line conductor is under maximum electrical stress and is likely to be punctured while the one nearest to the cross-arm is under minimum electrical stress.
2. The voltage distribution across various units depends upon the value of k and number of discs contained in the string. The greater the value of k, the more non-uniform is the voltage distribution across the discs and lesser is string efficiency. The inequality in voltage distribution increases with the number of discs in the string. Thus a shorter string has higher efficiency than the longer one.
When the insulators are wet the value of mutual capacitance C increases while C1 remains constant (except for the unit nearest the cross-arm) so the value of K decreases, more uniform potential distribution is obtained and the string efficiency increases.
The value of K (the ratio of shunt capacitance C1 to mutual capacitance C) varies and depends upon the length of the insulator string. The larger the number of insulator discs in a string, the longer will be the string. The longer the string, the greater must be the horizontal spacing between the insulator disc and the support (pole or tower) to make an allowance for conductor swing. The greater the horizontal spacing between the insulator string and the support, the lesser is the shunt capacitance C1 and vice-versa. Thus the value of K is low for longer strings and high for shorter strings. In practice K varies from 0.1 to 0.1667.