459 Appendix B: Stress and Capacitance Formulae In this appendix, formulae are derived for electric stress and capacitance for commonly existing electrode configurations in transformers such as two round electrodes or round electrode and plane. B1 Stress Calculations The information about the electric field intensity and potential field between two parallel cylindrical electrodes can be found by considering the respective equivalent line charges. Consider two line charges + ρ L and - ρ L (charges per unit length) placed at x=+m and x=-m respectively as shown in figure B1. Now, due to single line charge ρ L , the electric field intensity at a distance r is given by (B1) where e is permittivity of medium. The potential reckoned from a distance R is (B2) The resultant potential at point A (figure B1) due to line charges + ρ L and- ρ L is (B3) Copyright © 2004 by Marcel Dekker, Inc. Appendix B460 Let us now find the nature of equipotential surface having potential of u. From equation B3 we get But from figure B1 we have Solving by componendo and dividendo, By algebraic manipulations we get Figure B1 Two line charges placed at x=-m and x=+m Copyright © 2004 by Marcel Dekker, Inc. Appendix B 461 (B4) This is the equation of a circle with radius and center Thus, the equipotential surface is a cylinder which intersects the x-y plane in a circle with radius r and center at (s, 0). From the above expressions for radius and center we get (B5) (B6) By substituting the value of m in the equation for radius we have (B7) Now, from equations B5 and B7 we get Thus, we get the expression for potential as (B8) Now, we will consider two parallel cylindrical conductors of radii R 1 and R 2 , placed such that the distance between their centers is 2s. The electric field intensity and potential between the two conductors are calculated by considering the corresponding two equivalent line charges as shown in figure B2. Copyright © 2004 by Marcel Dekker, Inc. Appendix B462 Now, S 1 +S 2 =2s (B9) Using equation B6 we can write (B10) By solving equations B9 and B10 we get The electric field intensity at point P on the surface of the conductor on the right side is given by Now, by putting the value of in the above equation we get Figure B2 Configuration of two parallel cylindrical conductors Copyright © 2004 by Marcel Dekker, Inc. Appendix B 463 By putting the value of s 1 obtained earlier in the above equation we get (B11) Now, by using equation B8 for potential, the potential difference between points P and Q is given as By putting the values of s 1 and s 2 in the above equation and simplifying, Putting this value in equation for E p (equation B11) we have (B12) where Copyright © 2004 by Marcel Dekker, Inc. Appendix B464 is called as non-uniformity factor. Now, if both the electrodes have the same radius, i.e., R 1 =R 2 =R, then (B13) where Now, we will consider the other most commonly encountered geometry, i.e., cylindrical conductor—plane geometry as shown in figure B3. The ground plane at point G and the round conductor can be replaced by the configuration of the conductor and its image as shown in the figure. From equations B12 and B13 the electric field intensity, in this case, at point P is given as (B14) and non-uniformity factor is (B15) Figure B3 Cylindrical conductor—plane geometry Copyright © 2004 by Marcel Dekker, Inc. Appendix B 465 (B17) Putting this value in equation for E G we get (B18) (B19) The non-uniformity factor f x for any point x between the center of the conductor and ground in the x direction (figure B4) can now be found as below. The electric field intensity at a point with distance of x from the conductor center is (B20) Now, from equation B8 we have Now, the electric field intensity at point G is (B16) Using equation B8, the potential at the conductor surface is given as Copyright © 2004 by Marcel Dekker, Inc. Appendix B466 (B21) Putting this value in the equation for electric field we get (B22) (B23) The voltage at point x can be calculated using equation B3 as (B24) Figure B4 Stress and voltage at any point x from conductor center Copyright © 2004 by Marcel Dekker, Inc. Appendix B 467 (B25) B2 Capacitance Calculations B2.1 Capacitance between two parallel cylindrical conductors From figure B2 for the conditions that R 1 =R 2 =R and s 1 =s 2 =s, and by using equation B8, the capacitance between two parallel cylindrical conductors per unit length is given by (B26) Using equations B5 and B8 and simplifying we get the relation: (B27) From the equations B26 and B27, we finally get the capacitance per unit length as (B28) B2.2 Capacitance of cylindrical conductor and plane at ground potential From figure B4 and by using the equation B8, the capacitance per unit length between a conductor and ground plane is given by (B29) Copyright © 2004 by Marcel Dekker, Inc. Appendix B468 By using equation B27, we get the capacitance per unit length between the conductor and ground as (B30) Copyright © 2004 by Marcel Dekker, Inc. . points P and Q is given as By putting the values of s 1 and s 2 in the above equation and simplifying, Putting this value in equation for E p (equation B 11) we have (B 12) where Copyright © 20 04. © 20 04 by Marcel Dekker, Inc. Appendix B 467 (B25) B2 Capacitance Calculations B2 .1 Capacitance between two parallel cylindrical conductors From figure B2 for the conditions that R 1 =R 2 =R and. relation: (B27) From the equations B26 and B27, we finally get the capacitance per unit length as (B28) B2 .2 Capacitance of cylindrical conductor and plane at ground potential From figure B4 and by