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Generalized nonuniform transmission lines and their equivalent electrical length
Generalized
Nonun$orm
Transmission
and their Eguivalen t Electrical /ITA. G.J. HOLT,
and
K. u. AHMED
Lines
Length
P. BOWRON
Department of Electrical Engineering The University, flewcastle upon Tyne, U.K.
ABSTRACT
distributed tribution
:
The criterion
introduced
RC networks
is generalized
functions
network-properties
under each general at their
criterion is then illustrated for series-impedance for the exponential
by Hellstrom for obtaining to encompass
various classes of equivalent
a wide range of nonuniformity
cla.ss of transmission
lines which exhibit
equivalent
lengths.
respective
electrical
Application
by obtaining three generalized sets of nonuniformity
and shunt-admittance; of nonuniform
family
generalized set of nonuwzformity, derived from the generalized
and corresponding
some constant
hypergeometric
equivalent
lines. In order to demonstrate propagation
factor
clis-
identical of
the
distributions
electrical
lengths
the versatility
of a
distributions
are
line.
Introduction
The Telegrapher’s be written as
Equation
for any non-uniform
transmission
line may
(1) where 2 and the uniform variable and
=2(x) and Y = Y(x) represent the per-unit-length series-impedance per-unit-length shunt-admittance, respectively. Since for a non line, Z and Y are both considered as some functions of the distancex, they may a,lways be expressed as 2 = &f(x)
Pa)
Y = yog(x),
(2b)
where f(x) and g(x) may be designated as non-uniformity distribution functions with 2, being an impedance-constant and Y, an admittanceconstant. Acting on an original suggestion by Hildebrand (l),Holt and Ahmed (2, 3) recently transformed the Telegrapher’s Eq. 1 into the form
in terms of a new independent variable w = w(x), which is an arbitrary function of x. The coefficients of dV/dw and V may be put into t.he form
(44
31
A. G. J. Halt, K. U. Ahmed and P. Bowron and
(4b) and may be identified as the “classdiscriminant” for the general classification of nonuniform transmission lines (4). This generalized Telegrapher’s Eq. 3 and the “classdiscriminant” 4a, b have been used in deriving expressions for nonuniformity distribution functions f(x) and g(z) for certain general classes of transmission lines (3,4). Since the independent variable w(x) is, by hypothesis, an arbitrary function of x, Eq. 2 may easily be made to assume many of the specific forms available in previous work (5, 6, 7) as special cases of which only two instances of interest are cited below. Case (a) When
zy then w =
s
Iidw”4 1 dx
J(ZY)dx+a,
(54
’
= zc
(say),
(5b)
where a,, is any arbitrary constant. Substituting for w from Eq. 5b into Eq. 3, yields g-~[log,~(Z/Y)]~-k,
V = 0
(5c)
by virtue of
=&pi w%,JW
Y)l
= g- meJW VI. In Eq. 5c, k, is introduced as a frequency-factor (5) which depends upon the nature of the parameters distributed along the line. The transformed Eq. 5c was taken by Hellstrom (6) as the basis for formulating his criterion of equivalent distributed-parameter networks. Case (b) When
(64 then w=m[kdz+p)
32
=2,
(say),
(6b)
,Journal
of The Franklin
Institute
Generalized Transmission
Lines and their Equivalent Electrical Length
where constants m and p are chosen for consistency This further results in
with previous work (7).
Y ZY (dw/dx)2 = &% ’
(6~)
Thus, subject to the condition of (6a), Eq. 3 assumes the form
d2V ---
dv2
y
(64
v=-J
m2Z
’
of which the class-discriminant is given by the twin relations 6a, C. Equation 6d is employed by Bhattacharyya and Swamy (7) to obtain various interrelationships between Z(x) and Y(x) corresponding to the voltage-solutions of Eq. 6d as given in terms of certain special functions, such as Hermite, Whittakar functions, etc. Electrically
Equivalent
Nonuniforn
Lines
A network is defined as electrically equivalent (6) to another network when the electrical behavior of the Crst is identical with that of the second at their respective terminals, when subjected to the same operating conditions. In the case of distributed parameter networks, there may be an infinite number of pairs of the per-unit-length-distributed parameters &%,f&)~
yo,
%&)1,=1,2,
respectively, for Z,, Y,,f(x), g(x) of Eq. 2a, b where ZOn, Yo,d&), g,(x) stand, such that a network incorporating any of the nonuniformity pair, say, II: is e ec rically equivalent to a similar network characterized KJa(~)~ Y,cK&( )I 1 t by another nonuniformity pair, say, {Z,fa(x), YOJg,&)} so long as the equivalent electrical length of the first w, is equal to that of the second w8. This paper shows that any two pairs of nonuniformity parameters {Z,,f&), Y,, gn(x)},=1,2, ,,, having the equivalent network-property are directly obtainable from the generalized expressions of {2,&x), Y, g(x)} for one of the classes of generalized nonuniform lines treated earlier (24). In fact, the “class-discriminant” of Eqs. 4a, b constitutes the basis both for the generalized criterion of equivalent nonuniform lines considered here and for the general classification of nonuniform lines (4) with the exception that for its validity the former requires expression of equivalent electrical length corresponding to each set of nonuniformity pattern. Hellstrom gave his criterion (6) for deriving expressions for distributed parameters {%f(~), Y,g( x )> an d e q uivalent electrical length u, strictly on the basis of a comparison of the transformed Telegrapher’s Eq. 5c with any chosen standard linear second-order differential equation (L,DE), which may be manipulated into the following form: d2 V --P(zg-k, du2
v = 0,
analogous to Eq. 5c, where P(u) is a definite function
Vol.
288, No
1, July 1969
of u compatible
with
33
A. G. J. Halt, K. U. Ahmed and P. Bowron the solvability of the chosen standard L,DE. Since it is not possible to cast all known standard solvable L,DE’s into the form of Eq. 7, so as to render it appropriate for comparison with Eq. 5~3, the sphere of application of Hellstrom’s criterion is found to be restricted. However, the availability of the generalized Telegrapher’s L,DE in the form of Eq. (3), which is evidently more appropriate for comparison with any solvable standard L,DE, leads to (i) the general classification of nonuniform lines (4) and (ii) the generalization of Hellstrom’s criterion for equivalent nonuniform lines of the equivalent electrical length considered here. The general criterion for electrically equivalent nonuniform lines may be stated as follows : “The three sets of expressions, at most,, for the distributed parameters z(z) = +Q@(x) and Y(x) = Y,g( x ), one set consisting of Z(z) and Y(x) in terms of electrical length w(x), the second of Y(x) and W(X) in terms of arbitraryf(z), and the third of Z(x) and w(x) in terms of arbitrary g(x), are all obtainable from the ‘class-discriminant’ defined by the twin relations
and
2=
ZY
g P2(w). (8b) I( 1 “The ‘class-discriminant’ results directly from a comparison of the generalized Telegrapher’s Eq. 3 with a chosen standard L,DE, written in the form g-Pl(w)~-P2(w)
V = 0
PC)
having a well-known general solution, both equations being referred to the same independent variable w(x). “These three sets of expressions constitute the generalized pattern of nonuniformity for a class (associated with the name of the chosen standard L,DE meant for the comparison) of equivalent nonuniform lines. The equivalent electrical length indispensable for identical network-property is derived from the ‘class-discriminant’ Sa, b in terms of w(z), f(x), or g(z), as the case may be. The symbols PI(w) and P2(w) represent definite functions of the variable w compatible with the solvability of the standard L,DE chosen for comparison.” The application of the above criterion is illustrated below in deriving expressions for distributed parameters Z(Z) and Y(Z) and equivalent electrical length w for the family of generalized exponential lines. Example 1 For an exponential
line, the standard L,DE may be chosen as d2 V ~dw2
mi!I-V=(l rdw
(94
7
where m, is a constant of exponential-flare.
34
Journal of The Franklin
Institute
Generalized Transmission The class-discriminant
Lines and their Equivalent Electrical Length
for the exponential
A&Z 2 zy where
(I 1 I(dw”=l 1
line becomes
WI
=m,,
dx
(94
’
2 = -%Lf(X)~ y = yongw Equation
W W
(9c) gives w =
s
,/(ZY)dx+b,
= 4-G
5,)
s
J(fs) dx + b,,
W)
where b, is any arbitrary constant. Case (a) : When w = w(x) is any arbitrary function of x Manipulating
Eq. 9b as (d/d4 (Z/w’) (Z/w’)
and then integrating
= m w, 1
with respect to x, gives log, (Z/w’) = mI w + b,
where w’ = dw/dx and b, is any arbitrary constant. Taking antilogarithm leads to 2 = Z(x) = k, w’ emlw
(lOa)
where k, = eal is any constant ~0. Substituting for Z from Eq. 10a into Eq. 9c gives Y z Y(x) = (l/k,) w’ e-mlw. (lob) Thus, Eqs. lOa, b give one set of distributed parameters in terms of arbitrary equivalent electrical length w = w(x). Case (b): When f(x) = Z(x)/Z,, Substituting Eq. 9b gives
is an urbitrary
k&WWI which, on multiplying to 2 yields
288, No.
1,July 1969
=ml
both sides by Z and then integrating
M/Y) = ml
Vol.
function of x
the value of dw/dx in terms of Z and Y from Eq. 9c into
with respect
s
Zdx+c,
35
A. G. J. Halt, K. U. Ahmed and P. Bowron where co is any arbitrary
This may be simplified to give
constant. Y= Y(x) =
2
leading to
f(x)
Y(x) =
g(x) = y
0%
m,
4zo,
[
Y,,)
s
1
where c1 = coJ(Yon/Zon) is any constant. Substituting this value of g z g(x) into Eq. 9f gives
s
w = J(~onY,n)
ml
=
f(x)dx
s s 1
+b
0
d(Zon x3,) f dx + Cl
Alog, mlJVonY,J fh+cl [
Case (c): When g(x) = Y(x)/Y,,,
(114
2’
f dx+ cl
(lib)
+bo.
is an arbitrary function of x
Since
then substituting
this value in the relation
available in Case (b), gives 1 (d/d4 (z/Y) 22 JWY) which, on multiplying to x, becomes
= m,,
both sides by 2Y and then integrating
--&Y/Z)
with respect
= m,/Ydx-c,
where c2 is an arbitrary constant. This yields Z%z(x)
= [c2-mYjr Ydz ] 2
leading to f(x) = F!
where c3 = c, J(Zo,/Yo,)
36
=
2’ ~s-m&~o,Y,,) dx)dx s 1 [ g(x)
(124
is any constant.
Journal
of The Franklin
Institute
Generalized Transmission Inserting this value off(z)
Lines ad their Equivalent Electrical Length
into Eq. 9f gives
s
gdx
w = J(~o,Y,,) -
[
sI
cs--nz,J(z,nY,,)gdx +b,
-$%o[%-%J(Z,,%,) i c7bI+kl.
=
Cl=)
Speci$c Cases Let Z(x) = R(s),
Y(x) = C(Z), b, = 0, cl = cg = 1 and
(where p is a constant)
for Example
1. If, in addition,
(1) When G, = G, YO, = Cal, w = J(&,G_Jx, Eqs. lOa, b and 9f become
k, = J(&&,,,),
then
R(x) = Roleqx,
(100)
C(x) = Cole-qx,
w = J@&rJl)X. (2) Whenf(x)
= 1, Z,,
= Roz, Y,, = Co2, then Eqs. lla, b become
0
g(x) = -
= [~xJ&j+l]z.
(114
~loge[qxJ(~*j+l].
w =
\
J
(3) When g(z) = 1, Z,, = R,,, Y,, = Co3, then Eqs. 12a, b reduce to f(x)
=
w
=
Rc = _
[I-PJk&jj 23
J@Ol Cod
log
4
(124
.[l-qx JR&j] .i
These three specific cases lOc, llc, and 12c are found to agree with the results of Hellstrom (6), showing that his expressions are special cases of the corresponding generalized expressions lOa, b, lla, b, and 12a, b.
Constant
Propagation
Factor
Case
Lines with constant propagation tance for the practical fabrication
Vol. 258, No. 1, July
1968
factor r = ,/(ZY) are of particular imporof RC distributed networks in film form
37
A. G. J. Halt, K. U. Ahmed and P. Bowron and have been assumed in many previous treatments (8-11). In terms of the present nomenclature, such a condition requires that fg = &
0 0
a constant.
= k,,
(13)
Introducing a previously considered (2) class of generalized hypergeometric distributions as an example, the standard L,DE (Eq. Se) has - b(w - r3)
(I4a)
q(w) = (w-rl)(w-r2)’ 20xl
G(w) = (W-T-J
(14b)
(w-r2)j
where b is a coefficient constant, ri, r2, r3 are constants substituting Eqs. Sb and 14b, Eq. 13 becomes
w2 (w-rl)(w-r2)
=
and r,#r,.
Then
k2.
(15)
Taking the square root and integrating
where k, is a constant of integration which is omitted here since it only gives rise to a phase displacement. The result is (for the positive sign) r1r2 w = -----cosh2ax+---,
2
9”1+r2
(16)
2
where
The general impedance and admittance Eq. Sa and substitution of Eq. Sb are
functions found by integration
f = w’exp[ki(w)dw]
of
(I7a)
and I
pzw
g=w mexp
[i)?W dw] .
(17b)
Use of Eq. 14a in Eq. 17a gives f(x) = k,(w-r,)
b~~~z-~s~l~~l-~z~lS-f(w _ rl)b[(ra-rl)l(rl-r8)I-t,
(IS)
where k, is a constant introduced by the integration. After further substitution of Eq. 16 together with use of the hyperbolic half-angle properties Eq. 18 becomes f(x)
38
=
k4
(rr - r2)b-1
Journal
of The Franklin
Institute
Generalized Transmission
Lines and their Equivalent Electrical Length
Similarly, from Eq. 17b (19b) This distribution is seen to reduce to the earlier reported (11) raised hyperbolic sech power line when
b,;&$l 2
3
1
3
(204
and to the cosech power line when
b+$. Under the same conditions, by choosing trignometric cosec power line
POb)
the above
negative
integral,
the
Z(x) = 2, cosecn ax,
(214
Y(x) = Y, sin” ax,
(21b)
where n = 2(b - l), and the set power line emerge. The corresponding hypergeometric series parameters can then be derived from the generalized solutions to agree with those previously tabulated (11). Generalized
Conjluent
Hypergeometric
Lines
If, instead of Eqs. 14a and 14b, a degenerate equation is considered with Pi(W) =
form of the differential
-2,
(22a) 1
P2(w)
then using the substitution
=
4&l -__ W--T1
(22b)
w = - z + rl, the Telegrapher’s
d2 V ~~+[(rl-r3)--2]~+Z0Y0 This is a confluent hypergeometric V(z) =
Equation becomes
V = 0.
(23)
equation with the solution for all x of
C,,F,(% y; Z)+c~Z1--y&(l
fol--y,
2--y;
z),
(24)
where (y.= 2, Y,, y = rl - r3, C, and C, are constant. For identity with the general transformed differential Eq. 3, Eqs. 22a, b in Eqs. 17a, b yield (after integmtion) : - rJs--rl
(254
g(x) = g eW(w- rJT1-rs-l,
(25b)
f(x)
= k5 w’e+(w
and
where k, is an integration
Vol.
288, No. 1, July 1969
constant.
39
A. G. J. Halt, K. U. Ahmed and P. Bowron The constant propagation factor condition can again be applied as for the Gauss hypergeometric lines, this time Eqs. 17a, b and 22b give w = kxZ+rl, where k = P/4.&Y,,
(26)
so that Eqs. 25a, b become f(x) = (2k, krs+l+l e-h) g(x)
=
f
kQ-Ta-1
erl
x2(rs+d+l
x2(r1-rd-1
e-h’,
(27a)
e+kz*,
5
which describe a family of lines with weighted Gaussian distributions. Of particular interest is the simple Gaussian distribution which arises under the condition 2(r,-r,)+ 1 = 0 or r1-r3 = y = + (28) and can be shown (12) to be equivalent to an earlier solution (13) in terms of Hermite functions. Since fixed-diffusant impurity concentrations in semiconductors are known to follow a Gaussian Law (14), the solution of (24) above may be used to describe the distributed nature of some semiconductor monolithic structures. The equivalent electrical lengths for the two above constant propagation factors lines are given by Eqs. 16 and 26 with 2, and Y, the same in all cases. Conclusions
By transformation of the Telegrapher’s Eq. 1 a general class discriminant 8a, b is identified and used as a basis for deriving expressions for sets of generalized non-uniformity distribution, and their corresponding equivalent electrical lengths. Specific types of solutions previously reported are shown to be contained by the generalized form developed above, in particular that employed by Bhattacharyya and Swamy (7) to obtain interrelationships in terms of special functions. The approach also allows extension of Hellstrom’s equivalence criterion (6), and its application is illustrated by three types of generalized exponential line. Finally, it is shown that many known constant propagation factor lines of practical significance can be deduced from the general solutions for the Gauss and confluent hypergeometric cases. The concepts developed here for the classical transmission line can also be used in other devices, e.g. those employing plane-wave propagation in isotopic media characterized by TEM waves. Acknowledgements K. U. Ahmed wishes to acknowledge the financial support of the Government Assam (India) and P. Bowron that of the Electrosil Company Ltd., Sunderland.
of
References
(1) F. B. Hildebrand, “Advanced N. J., Prentice Hall, 1948.
40
Calculus for Engineers”,
p. 50. Englewood
Journal of The Frarklin
Cliffs,
Institute
Generalized Transmission
Lines and their Eqzcivalent Electrical Length
(2) A. G. J. Holt and K. U. Ahmed, “Generalized Distributions of Non-uniformity of Hypergeometric Transmission Lines”, Electronics Letters (IEE London), Vol. 4, pp. 167-168, May 1968. (3) A. G. J. Holt and K. U. Ahmed, “Generalized Patterns of Non-uniformity for Solvable Algebraic and Transcendental Transmission Lines”, Electron& Letters (IEE London), Vol. 4, pp. 287-289, July 1968. (4) A. G. J. Holt and K. U. Ahmed, “Exact Solutions of Generalized Non-uniform Lines and Their Classification”, J. Inst. Electronic and Radio Engineers, Vol. 36, pp. 373-379, Dec. 1968. RC and Lossless Transmission (5) K. J. Gough and R. N. Gould, “Non-uniform Lines” (correspondence) IEEE Trans. on Circuit Theory, Vol. CT-13, pp. 453454, Dec. 1966. (6) M. J. Hellstrom, “Equivalent Distributed RC Networks or Transmission Lines” IRE Trans. on Circuit Theory, Vol. CT-g, pp. 247-251, Sept. 1962. (7) B. B. Bhattacharyya and M. N. S. Swamy, “Some New RC lines with Special Functions for their Solutions”, J. Franklin Inst., Vol. 285, pp. 297-306, April 1968. Trans. (8) W. M. Kaufman and S. J. Garrett, “Tapered Distributed Networks”, IEEE, Vol. CT-g, pp. 329-336, 1962. (9) K. L. Su, “The Analysis of the Trigonometric RC line and some Applications”, Trans. IEEE, CT-II, pp. 158-159, 1964. (10) S. C. Dutta Roy, “Some Exactly Solvable Non-uniform RC Lines”, Trans. IEEE, CT-12, pp. 141-142, 1965. (11) A. G. J. Holt and P. Bowron, “Raised Trigonometric and Hyperbolic Power Lines”, Electronics Letters, Vol. 4, No. 7, pp. 140-142, April 1968. (12) A. G. J. Holt and P. Bowron, “Transformed Hypergeometric Transmission Lines”, Proc. IEE, Vol. 116, pp. 59-63, January 1969. (13) M. N. S. Swamy and B. B. Bhettacharyya, “Hermite Lines”, Proc. IEEE, Vol. 54, pp. 1577-78, 1966. (14) H. Lawrence and R. M. Warner, “Diffused J unction Depletion Layer Calculations”, Bell Systems Tech. J., Vol. 39, pp. 389-403, 1960.
Vol. 288. No. 1, July 1969
41
Nonun$orm
Transmission
and their Eguivalen t Electrical /ITA. G.J. HOLT,
and
K. u. AHMED
Lines
Length
P. BOWRON
Department of Electrical Engineering The University, flewcastle upon Tyne, U.K.
ABSTRACT
distributed tribution
:
The criterion
introduced
RC networks
is generalized
functions
network-properties
under each general at their
criterion is then illustrated for series-impedance for the exponential
by Hellstrom for obtaining to encompass
various classes of equivalent
a wide range of nonuniformity
cla.ss of transmission
lines which exhibit
equivalent
lengths.
respective
electrical
Application
by obtaining three generalized sets of nonuniformity
and shunt-admittance; of nonuniform
family
generalized set of nonuwzformity, derived from the generalized
and corresponding
some constant
hypergeometric
equivalent
lines. In order to demonstrate propagation
factor
clis-
identical of
the
distributions
electrical
lengths
the versatility
of a
distributions
are
line.
Introduction
The Telegrapher’s be written as
Equation
for any non-uniform
transmission
line may
(1) where 2 and the uniform variable and
=2(x) and Y = Y(x) represent the per-unit-length series-impedance per-unit-length shunt-admittance, respectively. Since for a non line, Z and Y are both considered as some functions of the distancex, they may a,lways be expressed as 2 = &f(x)
Pa)
Y = yog(x),
(2b)
where f(x) and g(x) may be designated as non-uniformity distribution functions with 2, being an impedance-constant and Y, an admittanceconstant. Acting on an original suggestion by Hildebrand (l),Holt and Ahmed (2, 3) recently transformed the Telegrapher’s Eq. 1 into the form
in terms of a new independent variable w = w(x), which is an arbitrary function of x. The coefficients of dV/dw and V may be put into t.he form
(44
31
A. G. J. Halt, K. U. Ahmed and P. Bowron and
(4b) and may be identified as the “classdiscriminant” for the general classification of nonuniform transmission lines (4). This generalized Telegrapher’s Eq. 3 and the “classdiscriminant” 4a, b have been used in deriving expressions for nonuniformity distribution functions f(x) and g(z) for certain general classes of transmission lines (3,4). Since the independent variable w(x) is, by hypothesis, an arbitrary function of x, Eq. 2 may easily be made to assume many of the specific forms available in previous work (5, 6, 7) as special cases of which only two instances of interest are cited below. Case (a) When
zy then w =
s
Iidw”4 1 dx
J(ZY)dx+a,
(54
’
= zc
(say),
(5b)
where a,, is any arbitrary constant. Substituting for w from Eq. 5b into Eq. 3, yields g-~[log,~(Z/Y)]~-k,
V = 0
(5c)
by virtue of
=&pi w%,JW
Y)l
= g- meJW VI. In Eq. 5c, k, is introduced as a frequency-factor (5) which depends upon the nature of the parameters distributed along the line. The transformed Eq. 5c was taken by Hellstrom (6) as the basis for formulating his criterion of equivalent distributed-parameter networks. Case (b) When
(64 then w=m[kdz+p)
32
=2,
(say),
(6b)
,Journal
of The Franklin
Institute
Generalized Transmission
Lines and their Equivalent Electrical Length
where constants m and p are chosen for consistency This further results in
with previous work (7).
Y ZY (dw/dx)2 = &% ’
(6~)
Thus, subject to the condition of (6a), Eq. 3 assumes the form
d2V ---
dv2
y
(64
v=-J
m2Z
’
of which the class-discriminant is given by the twin relations 6a, C. Equation 6d is employed by Bhattacharyya and Swamy (7) to obtain various interrelationships between Z(x) and Y(x) corresponding to the voltage-solutions of Eq. 6d as given in terms of certain special functions, such as Hermite, Whittakar functions, etc. Electrically
Equivalent
Nonuniforn
Lines
A network is defined as electrically equivalent (6) to another network when the electrical behavior of the Crst is identical with that of the second at their respective terminals, when subjected to the same operating conditions. In the case of distributed parameter networks, there may be an infinite number of pairs of the per-unit-length-distributed parameters &%,f&)~
yo,
%&)1,=1,2,
respectively, for Z,, Y,,f(x), g(x) of Eq. 2a, b where ZOn, Yo,d&), g,(x) stand, such that a network incorporating any of the nonuniformity pair, say, II: is e ec rically equivalent to a similar network characterized KJa(~)~ Y,cK&( )I 1 t by another nonuniformity pair, say, {Z,fa(x), YOJg,&)} so long as the equivalent electrical length of the first w, is equal to that of the second w8. This paper shows that any two pairs of nonuniformity parameters {Z,,f&), Y,, gn(x)},=1,2, ,,, having the equivalent network-property are directly obtainable from the generalized expressions of {2,&x), Y, g(x)} for one of the classes of generalized nonuniform lines treated earlier (24). In fact, the “class-discriminant” of Eqs. 4a, b constitutes the basis both for the generalized criterion of equivalent nonuniform lines considered here and for the general classification of nonuniform lines (4) with the exception that for its validity the former requires expression of equivalent electrical length corresponding to each set of nonuniformity pattern. Hellstrom gave his criterion (6) for deriving expressions for distributed parameters {%f(~), Y,g( x )> an d e q uivalent electrical length u, strictly on the basis of a comparison of the transformed Telegrapher’s Eq. 5c with any chosen standard linear second-order differential equation (L,DE), which may be manipulated into the following form: d2 V --P(zg-k, du2
v = 0,
analogous to Eq. 5c, where P(u) is a definite function
Vol.
288, No
1, July 1969
of u compatible
with
33
A. G. J. Halt, K. U. Ahmed and P. Bowron the solvability of the chosen standard L,DE. Since it is not possible to cast all known standard solvable L,DE’s into the form of Eq. 7, so as to render it appropriate for comparison with Eq. 5~3, the sphere of application of Hellstrom’s criterion is found to be restricted. However, the availability of the generalized Telegrapher’s L,DE in the form of Eq. (3), which is evidently more appropriate for comparison with any solvable standard L,DE, leads to (i) the general classification of nonuniform lines (4) and (ii) the generalization of Hellstrom’s criterion for equivalent nonuniform lines of the equivalent electrical length considered here. The general criterion for electrically equivalent nonuniform lines may be stated as follows : “The three sets of expressions, at most,, for the distributed parameters z(z) = +Q@(x) and Y(x) = Y,g( x ), one set consisting of Z(z) and Y(x) in terms of electrical length w(x), the second of Y(x) and W(X) in terms of arbitraryf(z), and the third of Z(x) and w(x) in terms of arbitrary g(x), are all obtainable from the ‘class-discriminant’ defined by the twin relations
and
2=
ZY
g P2(w). (8b) I( 1 “The ‘class-discriminant’ results directly from a comparison of the generalized Telegrapher’s Eq. 3 with a chosen standard L,DE, written in the form g-Pl(w)~-P2(w)
V = 0
PC)
having a well-known general solution, both equations being referred to the same independent variable w(x). “These three sets of expressions constitute the generalized pattern of nonuniformity for a class (associated with the name of the chosen standard L,DE meant for the comparison) of equivalent nonuniform lines. The equivalent electrical length indispensable for identical network-property is derived from the ‘class-discriminant’ Sa, b in terms of w(z), f(x), or g(z), as the case may be. The symbols PI(w) and P2(w) represent definite functions of the variable w compatible with the solvability of the standard L,DE chosen for comparison.” The application of the above criterion is illustrated below in deriving expressions for distributed parameters Z(Z) and Y(Z) and equivalent electrical length w for the family of generalized exponential lines. Example 1 For an exponential
line, the standard L,DE may be chosen as d2 V ~dw2
mi!I-V=(l rdw
(94
7
where m, is a constant of exponential-flare.
34
Journal of The Franklin
Institute
Generalized Transmission The class-discriminant
Lines and their Equivalent Electrical Length
for the exponential
A&Z 2 zy where
(I 1 I(dw”=l 1
line becomes
WI
=m,,
dx
(94
’
2 = -%Lf(X)~ y = yongw Equation
W W
(9c) gives w =
s
,/(ZY)dx+b,
= 4-G
5,)
s
J(fs) dx + b,,
W)
where b, is any arbitrary constant. Case (a) : When w = w(x) is any arbitrary function of x Manipulating
Eq. 9b as (d/d4 (Z/w’) (Z/w’)
and then integrating
= m w, 1
with respect to x, gives log, (Z/w’) = mI w + b,
where w’ = dw/dx and b, is any arbitrary constant. Taking antilogarithm leads to 2 = Z(x) = k, w’ emlw
(lOa)
where k, = eal is any constant ~0. Substituting for Z from Eq. 10a into Eq. 9c gives Y z Y(x) = (l/k,) w’ e-mlw. (lob) Thus, Eqs. lOa, b give one set of distributed parameters in terms of arbitrary equivalent electrical length w = w(x). Case (b): When f(x) = Z(x)/Z,, Substituting Eq. 9b gives
is an urbitrary
k&WWI which, on multiplying to 2 yields
288, No.
1,July 1969
=ml
both sides by Z and then integrating
M/Y) = ml
Vol.
function of x
the value of dw/dx in terms of Z and Y from Eq. 9c into
with respect
s
Zdx+c,
35
A. G. J. Halt, K. U. Ahmed and P. Bowron where co is any arbitrary
This may be simplified to give
constant. Y= Y(x) =
2
leading to
f(x)
Y(x) =
g(x) = y
0%
m,
4zo,
[
Y,,)
s
1
where c1 = coJ(Yon/Zon) is any constant. Substituting this value of g z g(x) into Eq. 9f gives
s
w = J(~onY,n)
ml
=
f(x)dx
s s 1
+b
0
d(Zon x3,) f dx + Cl
Alog, mlJVonY,J fh+cl [
Case (c): When g(x) = Y(x)/Y,,,
(114
2’
f dx+ cl
(lib)
+bo.
is an arbitrary function of x
Since
then substituting
this value in the relation
available in Case (b), gives 1 (d/d4 (z/Y) 22 JWY) which, on multiplying to x, becomes
= m,,
both sides by 2Y and then integrating
--&Y/Z)
with respect
= m,/Ydx-c,
where c2 is an arbitrary constant. This yields Z%z(x)
= [c2-mYjr Ydz ] 2
leading to f(x) = F!
where c3 = c, J(Zo,/Yo,)
36
=
2’ ~s-m&~o,Y,,) dx)dx s 1 [ g(x)
(124
is any constant.
Journal
of The Franklin
Institute
Generalized Transmission Inserting this value off(z)
Lines ad their Equivalent Electrical Length
into Eq. 9f gives
s
gdx
w = J(~o,Y,,) -
[
sI
cs--nz,J(z,nY,,)gdx +b,
-$%o[%-%J(Z,,%,) i c7bI+kl.
=
Cl=)
Speci$c Cases Let Z(x) = R(s),
Y(x) = C(Z), b, = 0, cl = cg = 1 and
(where p is a constant)
for Example
1. If, in addition,
(1) When G, = G, YO, = Cal, w = J(&,G_Jx, Eqs. lOa, b and 9f become
k, = J(&&,,,),
then
R(x) = Roleqx,
(100)
C(x) = Cole-qx,
w = J@&rJl)X. (2) Whenf(x)
= 1, Z,,
= Roz, Y,, = Co2, then Eqs. lla, b become
0
g(x) = -
= [~xJ&j+l]z.
(114
~loge[qxJ(~*j+l].
w =
\
J
(3) When g(z) = 1, Z,, = R,,, Y,, = Co3, then Eqs. 12a, b reduce to f(x)
=
w
=
Rc = _
[I-PJk&jj 23
J@Ol Cod
log
4
(124
.[l-qx JR&j] .i
These three specific cases lOc, llc, and 12c are found to agree with the results of Hellstrom (6), showing that his expressions are special cases of the corresponding generalized expressions lOa, b, lla, b, and 12a, b.
Constant
Propagation
Factor
Case
Lines with constant propagation tance for the practical fabrication
Vol. 258, No. 1, July
1968
factor r = ,/(ZY) are of particular imporof RC distributed networks in film form
37
A. G. J. Halt, K. U. Ahmed and P. Bowron and have been assumed in many previous treatments (8-11). In terms of the present nomenclature, such a condition requires that fg = &
0 0
a constant.
= k,,
(13)
Introducing a previously considered (2) class of generalized hypergeometric distributions as an example, the standard L,DE (Eq. Se) has - b(w - r3)
(I4a)
q(w) = (w-rl)(w-r2)’ 20xl
G(w) = (W-T-J
(14b)
(w-r2)j
where b is a coefficient constant, ri, r2, r3 are constants substituting Eqs. Sb and 14b, Eq. 13 becomes
w2 (w-rl)(w-r2)
=
and r,#r,.
Then
k2.
(15)
Taking the square root and integrating
where k, is a constant of integration which is omitted here since it only gives rise to a phase displacement. The result is (for the positive sign) r1r2 w = -----cosh2ax+---,
2
9”1+r2
(16)
2
where
The general impedance and admittance Eq. Sa and substitution of Eq. Sb are
functions found by integration
f = w’exp[ki(w)dw]
of
(I7a)
and I
pzw
g=w mexp
[i)?W dw] .
(17b)
Use of Eq. 14a in Eq. 17a gives f(x) = k,(w-r,)
b~~~z-~s~l~~l-~z~lS-f(w _ rl)b[(ra-rl)l(rl-r8)I-t,
(IS)
where k, is a constant introduced by the integration. After further substitution of Eq. 16 together with use of the hyperbolic half-angle properties Eq. 18 becomes f(x)
38
=
k4
(rr - r2)b-1
Journal
of The Franklin
Institute
Generalized Transmission
Lines and their Equivalent Electrical Length
Similarly, from Eq. 17b (19b) This distribution is seen to reduce to the earlier reported (11) raised hyperbolic sech power line when
b,;&$l 2
3
1
3
(204
and to the cosech power line when
b+$. Under the same conditions, by choosing trignometric cosec power line
POb)
the above
negative
integral,
the
Z(x) = 2, cosecn ax,
(214
Y(x) = Y, sin” ax,
(21b)
where n = 2(b - l), and the set power line emerge. The corresponding hypergeometric series parameters can then be derived from the generalized solutions to agree with those previously tabulated (11). Generalized
Conjluent
Hypergeometric
Lines
If, instead of Eqs. 14a and 14b, a degenerate equation is considered with Pi(W) =
form of the differential
-2,
(22a) 1
P2(w)
then using the substitution
=
4&l -__ W--T1
(22b)
w = - z + rl, the Telegrapher’s
d2 V ~~+[(rl-r3)--2]~+Z0Y0 This is a confluent hypergeometric V(z) =
Equation becomes
V = 0.
(23)
equation with the solution for all x of
C,,F,(% y; Z)+c~Z1--y&(l
fol--y,
2--y;
z),
(24)
where (y.= 2, Y,, y = rl - r3, C, and C, are constant. For identity with the general transformed differential Eq. 3, Eqs. 22a, b in Eqs. 17a, b yield (after integmtion) : - rJs--rl
(254
g(x) = g eW(w- rJT1-rs-l,
(25b)
f(x)
= k5 w’e+(w
and
where k, is an integration
Vol.
288, No. 1, July 1969
constant.
39
A. G. J. Halt, K. U. Ahmed and P. Bowron The constant propagation factor condition can again be applied as for the Gauss hypergeometric lines, this time Eqs. 17a, b and 22b give w = kxZ+rl, where k = P/4.&Y,,
(26)
so that Eqs. 25a, b become f(x) = (2k, krs+l+l e-h) g(x)
=
f
kQ-Ta-1
erl
x2(rs+d+l
x2(r1-rd-1
e-h’,
(27a)
e+kz*,
5
which describe a family of lines with weighted Gaussian distributions. Of particular interest is the simple Gaussian distribution which arises under the condition 2(r,-r,)+ 1 = 0 or r1-r3 = y = + (28) and can be shown (12) to be equivalent to an earlier solution (13) in terms of Hermite functions. Since fixed-diffusant impurity concentrations in semiconductors are known to follow a Gaussian Law (14), the solution of (24) above may be used to describe the distributed nature of some semiconductor monolithic structures. The equivalent electrical lengths for the two above constant propagation factors lines are given by Eqs. 16 and 26 with 2, and Y, the same in all cases. Conclusions
By transformation of the Telegrapher’s Eq. 1 a general class discriminant 8a, b is identified and used as a basis for deriving expressions for sets of generalized non-uniformity distribution, and their corresponding equivalent electrical lengths. Specific types of solutions previously reported are shown to be contained by the generalized form developed above, in particular that employed by Bhattacharyya and Swamy (7) to obtain interrelationships in terms of special functions. The approach also allows extension of Hellstrom’s equivalence criterion (6), and its application is illustrated by three types of generalized exponential line. Finally, it is shown that many known constant propagation factor lines of practical significance can be deduced from the general solutions for the Gauss and confluent hypergeometric cases. The concepts developed here for the classical transmission line can also be used in other devices, e.g. those employing plane-wave propagation in isotopic media characterized by TEM waves. Acknowledgements K. U. Ahmed wishes to acknowledge the financial support of the Government Assam (India) and P. Bowron that of the Electrosil Company Ltd., Sunderland.
of
References
(1) F. B. Hildebrand, “Advanced N. J., Prentice Hall, 1948.
40
Calculus for Engineers”,
p. 50. Englewood
Journal of The Frarklin
Cliffs,
Institute
Generalized Transmission
Lines and their Eqzcivalent Electrical Length
(2) A. G. J. Holt and K. U. Ahmed, “Generalized Distributions of Non-uniformity of Hypergeometric Transmission Lines”, Electronics Letters (IEE London), Vol. 4, pp. 167-168, May 1968. (3) A. G. J. Holt and K. U. Ahmed, “Generalized Patterns of Non-uniformity for Solvable Algebraic and Transcendental Transmission Lines”, Electron& Letters (IEE London), Vol. 4, pp. 287-289, July 1968. (4) A. G. J. Holt and K. U. Ahmed, “Exact Solutions of Generalized Non-uniform Lines and Their Classification”, J. Inst. Electronic and Radio Engineers, Vol. 36, pp. 373-379, Dec. 1968. RC and Lossless Transmission (5) K. J. Gough and R. N. Gould, “Non-uniform Lines” (correspondence) IEEE Trans. on Circuit Theory, Vol. CT-13, pp. 453454, Dec. 1966. (6) M. J. Hellstrom, “Equivalent Distributed RC Networks or Transmission Lines” IRE Trans. on Circuit Theory, Vol. CT-g, pp. 247-251, Sept. 1962. (7) B. B. Bhattacharyya and M. N. S. Swamy, “Some New RC lines with Special Functions for their Solutions”, J. Franklin Inst., Vol. 285, pp. 297-306, April 1968. Trans. (8) W. M. Kaufman and S. J. Garrett, “Tapered Distributed Networks”, IEEE, Vol. CT-g, pp. 329-336, 1962. (9) K. L. Su, “The Analysis of the Trigonometric RC line and some Applications”, Trans. IEEE, CT-II, pp. 158-159, 1964. (10) S. C. Dutta Roy, “Some Exactly Solvable Non-uniform RC Lines”, Trans. IEEE, CT-12, pp. 141-142, 1965. (11) A. G. J. Holt and P. Bowron, “Raised Trigonometric and Hyperbolic Power Lines”, Electronics Letters, Vol. 4, No. 7, pp. 140-142, April 1968. (12) A. G. J. Holt and P. Bowron, “Transformed Hypergeometric Transmission Lines”, Proc. IEE, Vol. 116, pp. 59-63, January 1969. (13) M. N. S. Swamy and B. B. Bhettacharyya, “Hermite Lines”, Proc. IEEE, Vol. 54, pp. 1577-78, 1966. (14) H. Lawrence and R. M. Warner, “Diffused J unction Depletion Layer Calculations”, Bell Systems Tech. J., Vol. 39, pp. 389-403, 1960.
Vol. 288. No. 1, July 1969
41