Let the capacitance of each insulator be C and the capacitance between each pin and earth be Cp. Then we know that the self-capacitance of the string is 4C and the total pin to earth capacitance is 4Cp.
The voltage distribution across each unit is given by the formula:
Vn = Vt * ((Cp + nC) / (4Cp + 4nC))
where Vn is the voltage across the nth unit, Vt is the total voltage across the string, and n is the unit number (n = 1 for the first unit, n = 2 for the second unit, and so on).
Substituting the values we have:
V1 = Vt * ((Cp + C) / (4Cp + 4C))
V2 = Vt * ((Cp + 2C) / (4Cp + 8C))
V3 = Vt * ((Cp + 3C) / (4Cp + 12C))
V4 = Vt * ((Cp + 4C) / (4Cp + 16C))
We can simplify this expression by dividing both numerator and denominator by 4Cp:
Vn = Vt * ((1/4) + (n/4) * (C/Cp)) / (1 + n * (C/Cp))
Now we can calculate the percentage voltage distribution across each unit:
V1/Vt = ((1/4) + (1/4) * (C/Cp)) / (1 + 1 * (C/Cp))
V2/Vt = ((1/4) + (2/4) * (C/Cp)) / (1 + 2 * (C/Cp))
V3/Vt = ((1/4) + (3/4) * (C/Cp)) / (1 + 3 * (C/Cp))
V4/Vt = ((1/4) + (4/4) * (C/Cp)) / (1 + 4 * (C/Cp))
We can simplify these expressions further to get:
V1/Vt = (1 / (5 + C/Cp)) * 100%
V2/Vt = (2 / (9 + 2C/Cp)) * 100%
V3/Vt = (3 / (13 + 3C/Cp)) * 100%
V4/Vt = (4 / (17 + 4C/Cp)) * 100%
Now we can calculate the string efficiency as the ratio of voltage across the string to the voltage across the equivalent capacitance of the insulators:
String efficiency = Vt / (4C)
= ((Cp + C) / (4Cp + 4C)) + ((Cp + 2C) / (4Cp + 8C)) + ((Cp + 3C) / (4Cp + 12C)) + ((Cp + 4C) / (4Cp + 16C))
= (5Cp + 10C) / (4Cp + 16C)
= (5/4) * (Cp/C + 2) / (1 + 4Cp/C)
