Potential for Cylindrical Problems
7.2 APPLICATION OF A METHOD OF COMPLEX POTENTIAL TO PIECEWISE HOMOGENEOUS MEDIA
- +
2 . (7.12)
In this and follow chapters the method of complex potential is generalized, firstly, for piecewise homogeneous media and, secondly, for three-dimensional structures.
7.2 APPLICATION OF A METHOD OF COMPLEX POTENTIAL TO PIECEWISE HOMOGENEOUS MEDIA
The problem of a field calculation becomes complicated in the case of the heterogeneous (in particular, piecewise-homogeneous) medium [64]. Such problem arises, if the solitary wire is located at the boundaries of the two media, for example, air and dielectric, and the relative permittivity er of dielectric is different from 1 (Figure 7.2). The wire cross-section is considered as a circle.
It should be noted that the potential at the wire surface is constant. The potential of the infinitely far point may not be equal to zero, since in that case the potential of infinitely long wires at the all finite distances will be infinitely great. Nevertheless it is obvious that at the great distances from charged wire the potential in the all directions is the same. And therefore suppositions that the lines of equal potential are circumferences with the center in the origin of coordinates and that the surfaces of equal potential are the surfaces of the circular cylinders are natural.
Figure 7.2 Field of a wire located at two media boundary.
The wire surface may be replaced by a system of N equidistant filaments with linear densities tnof charges (n is the filament number), the sum of which is equal to t (to the linear density of wire charge). The potential of each filament at distance r from its axis is
Un = –(tn/2p en)ln (r/rp). (7.13) Here, en is the permittivity of a medium around the filament, rp is the distance to the surface of zero potential. In order that the potentials of all filaments may be identical on the surface of circular cylinder with the radius r = r + a, where a is the wire radius, the ratio tn/en must be the same for all the n:
tn/en=const(n), (7.14)
i.e. the surface density of the wire charge must be proportional to the permittivity of a medium adjacent to a given part of the surface. Here the word “adjacent” signifies that the medium occupies volume from the wire surface to infinity within an angle equal to the arc length of the surface part, which is adjacent to this medium.
Formula (7.11) describing a field structure of the solitary wire with the circular cross-section, which is located in homogeneous medium, may be written in the form of a more general expression:
V(z) = Aj ln z + C, (7.15)
where A is constant magnitude. Using designation z = rejj, we obtain:
V(z) = –Aj + C1, U(z) = A ln r + C2, (7.16) from whence the equations of field lines and of equal potential lines take the form accordingly:
j = const, r = const. (7.17)
The field structure of the wire located at the boundary of the two media has a similar character. In particular, as it is indicated earlier, here the lines of equal potential are circumferences with r = const. Therefore for the complex potential of such field we shall use the expression (7.15).
The constant A in this expression is determined in accord with the fact that in going around the cross-section of the wire along the shorted contour, the angle j increases by 2p, and the function V increases by YE/l, where YE is the flux of the vector E through the cylindrical surface covering the wire segment of length l. In the case of
the heterogeneous medium it is expedient to replace YE by YD, i.e. to replace the flux of vectorE
by the flux of vectorD
, where D
is the vector of electrical displacement. Besides, one must take into account that the strength of this flux in the different media may be diverse, i.e.
DV Y
l i Di i
=1
Â
/e, (7.18)where i is the medium number.
If rv is a volume density of a wire charge, then integration of Maxwell equation divD V
= r , (7.19)
with respect to wire volume v gives for left part of equation divDdv DdS DdSi density of a surface charge on this area. Equating both sides of the equation, we find
YDi i
i i
=
Â
qÂ
, (7.20)i.e., the flux YDi of the displacement vector through the wire surface into i-medium is equal to the charge qi per unit length of the surface area adjacent to this medium.
In accordance with (7.18)
DV l qi i
i
=1
Â
/e. (7.21)If to introduce such a quantity eethat
qi i q e
i
/e = /e
Â
, (7.22)where q is the total wire charge per unit of its length, then as one can see from (7.22), the quantity eehas meaning of the equivalent permittivity of the heterogeneous medium.
Accordingly (7.14), if Dji is the arc length of the wire circumference, which is adjacent to the i-medium, then the equality
qi/(e jiD i)=const i( ), (7.23) is true. It is obvious also that
qi q
i
Â
= . (7.24)It is easy to show by using (7.23) and (7.24) that
If N media of the same angle width are adjacent to the wire, then
ee ei
For variant, depicted in Figure 7.2, N = 2, and the equivalent permittivity is equal to the arithmetic average of magnitudes e1 and e2:
ee =(e1+e2)/2. (7.28)
Substituting (7.22) into (7.21) and taking into account that according to (7.16) in going around the cross-section of wire along the shorted contour the function V increases by It is necessary to note that only the magnitude eeis included in the equalities (7.31).
That means that in this case, as in the case of the homogeneous medium, the angles between the lines of the fields are equal to each other (irrespective of the medium, in which they are located), if the increase of a flux at the transition from one line of the field to other line is identical. If increase of potential between neighbors lines of the equal potential is also the same, the radii of the circumferences of the equal potential change according to geometric progression.
Consider using the obtained results, the important practical case of a two-wire line (Figure 7.3). In the beginning we shall assume that the wires are infinitely thin.
The expression (7.12) for the complex potential of the field of such a line located in a homogenous medium may be rewritten in the form
V( )z Ajlnz b z b C
= +
- + . (7.32)
If designate z + b = r1 exp(jj1), z – b = r2 exp(jj2), where r1and r2are the lengths of the segments connecting the observation point M with the axes of the wires, and j1and j2are the angles between these segments and the axis x, we find
V=A(j2-j1), U=Aln(r1/r2). (7.33) Here it is accepted that C = 0. In this case the line of zero potential is the ordinate axis, and the sections of abscissa axis from the wires of the line to infinity. The lines of equal
potential are the circumferences with the centers on the axis x, and the lines of the field are the circumferences, passing through the wires axes, with the centers on the axis y.
Let a circular dielectric cylinder of radius b be placed between two wires of line, and its axis is parallel to the wires and is located at the same distances from both wires. The dielectric boundary (the circumference) coincides with the line of the field. The lines of equal potential intersect it at the right angle. It means that the dielectric cylinder doesn’t change the field structure.
Figure 7.3 Field of two wires located along the generatrices of the dielectric cylinder.
As in the case of the solitary wire, in order to the potential near the wire was the identical in the air and in the dielectric, the surface density of the charge must be proportional to the permittivity of the medium, adjacent to a given part of the wires. So the magnitude A in the expressions (7.32)–(7.33) is defined by the equality (7.30), and the equivalent permittivity is defined by the expression (7.28).
If the wires of the real transmission line are not infinitely thin, but have circular cross-sections of the finite radius, then always one can place the axes of the wires such that the surfaces of the real wires may coincide with the surfaces of equal potential, which are the surfaces of the round cylinders.
The two-wire line may be fabricated in the shape of the round metallic cylinder located over the metal plane (Figure 7.4a), in the shape of two round cylinders with different axes, which do not encompass one another (Figure 7.4b) and in the shape of two analogous cylinders encompassing one another (Figure 7.4c). The dielectric cylinder is used in all these cases simultaneously both as an isolator and as a supporting construction. One must note that the circumference coinciding with the boundary of the dielectric cylinder goes not through axes of the round metallic cylinders, but through equivalent infinitely thin straight wires, the fields of which coincide with the fields of the round cylinders. Two equipotential surfaces of fields of thin wires coincide with the surfaces of the real wires.
The distance between the axis of the round metallic cylinder and the equivalent wire for the variant shown in Figure 7.4a is equal to
h b- = -h h2-R2, (7.34) where R is the radius of the round cylinder. For the variant shown in Figure 7.4b the distances between the surface of zero potential and the axis of one round cylinder and between this surface and the equivalent wire are equal correspondingly to
h D R R
D b h R
D D R R D R R
1
2 12 22
12 12
1 2 22
1 2 22
2
1
= + - , = - =2 ÈÎ( - ) - ˘˚ÈÎ( + ) - ˘˚˚, (7.35) where R1 and R2 are the radii of the cylinders, and D is the distance between their axes.
At last for the variant depicted in Figure 7.4c
h R D R
D b h R
1 22 2
12
12 12
= -2 - , = - . (7.36)
Figure 7.4 The long line in the shape of the round metallic cylinder over the metal plane (a) and in the shape of two cylinders with different axes, which do not encompass (b) and encompass (c) one another.
The capacitances of the solitary wire and of the two-wire transmission line are proportional to the permittivity e of medium. In the case of the heterogeneous medium consisting of homogeneous layers the magnitude e must be replaced by the equivalent permittivity—in accordance with above-presented expressions. In particular, if the solitary wire is located at the boundary of the two media (see Figure 7.2), the equivalent permittivity is equal to the arithmetic average of the magnitudes e1 and e2. This conclusion conforms completely to the known thesis, in accordance with which the capacitance between two conductors located symmetrically relative to a flat boundary of two media with the permittivity e1 and e2is equal to half-sum of the capacitance values between the same conductors in the homogeneous media with permittivity e1 and e2
accordingly [34]. In the case of a single wire, the metal cylinder of infinite radius, axis of which coincides with the wire axis, may be accepted as the second conductor.
If the boundary of the two media goes along a broken line, whose point of sharp bend coincides with the wire center (this boundary is shown by dotted line in Figure 7.2), then in accordance with (7.26)
ee = 1p e j +e j
2 ( 1D 1 2D 2). (7.37)
If a few media, the boundaries of which coincide with the radial surfaces, are adjacent to the solitary wire, then in the case of equal angular width of the adjacent media, the equivalent permittivity is determined by the expression (7.27), in the case of different angular width this permittivity is determined by (7.26). This conclusion coincides with the results obtained in [65, 66].
In [65] it is shown that the electrostatic field of the wires’ system of the arbitrary shape located in a piecewise homogenous medium coincides with a field in a homogenous medium, if the media boundaries coincide with the surfaces of wires and the surfaces of the equal strength of the field in the homogenous medium (this condition is called the condition of invariance). Accordingly for the capacitance between the wires located in a piecewise homogenous medium [66] gives the expression
Here Ci0 is the capacitance between the medium i and the wire segment adjacent to it, if the wire is located in a homogeneous medium with permittivity e0.
It is obvious that the condition of invariance is performed in the case when the boundaries of the media adjacent to the solitary wire coincide with radial surfaces. If to use as the second wire the metal cylinder of the infinite radius coaxial with the solitary wire, then it is easy to obtain (7.26) and (7.27) from (7.38).
For the line from two infinitely thin wires located along the generatrices of the circular dielectric cylinder, the condition of invariance is performed also. Since the angle width of both media adjacent to each wire is equal to Dj1 = Dj2 = p, then C10 = C20, i.e.
the magnitude ee, is also determined by the equality (7.28). This proposition stays true, if the thin wires are located along the arbitrary selected generatrices, the length of arc between which is not equal to p.
If a thin dielectric insert with the angle width 2a, limited by arcs of two circumferences, the centers of which lie on the axis y in the points y = ±d, is placed between the wires (the insert boundaries are shown by the dotted lines in Figure7.3), then in accordance with (7.26)
ee = e1 +(e2 – e1) a/p, (7.39) where e1 is the permittivity of the air and e2 is the permittivity of the dielectric.
The capacitance between the metallic cylinders per unit length (see Figure 7.4) is C
where the plus sign in the second square brackets is taken when one must calculate the capacitance between the cylinders depicted in Figure 7.4b, and the minus sign corresponds to Figure 7.4c.
For two cylinders of the same radius (R1 = R2 = R)
This expression coincides with the expression, which one may obtain from presented in [34] expression (8–7) for the linear capacitance of the two-wire line located in the dielectric near the cylindrical interface of the two media, if the distance between the axis of the each wire and the interface is a small magnitude of the order of the wire radius. One must note that the expression (8–7) and the expression (7.41) are true at the arbitrary location of the charged filaments (of the equivalent linear wires) along the cylindrical interface of two media, i.e. at any arc length between the filaments (not only at arc length p).
For the circular cylinder located over a metal plane (see Figure 7.4a) C
In the particular case, if h/R 1,
C= 2pe /ln( / ),e 2h R (7.43) that coincides with the expression, which one may obtain from presented in [34]
expression (8–8) for the two-wire line with isolation of the ribbon type.