transformer engineering design and practice 1_phần 2 docx

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 equiv

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 +ρL and -ρL (charges per unit

length) placed at x=+m and x=-m respectively as shown in figure B1 Now, due to single line charge ρL , 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 +ρL and-ρL is

(B3)

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

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 R1 and R2, 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

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

Appendix B 463

By putting the value of s1 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 s1 and s2 in the above equation and simplifying,

Putting this value in equation for E p (equation B11) we have

(B12) where

Now, if both the electrodes have the same radius, i.e., R1=R2=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

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

(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

Appendix B 467

(B25)

B2 Capacitance Calculations

B2.1 Capacitance between two parallel cylindrical conductors

From figure B2 for the conditions that R1=R2=R and s1=s2=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)

conductor and ground as

(B30)