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210
Approximation of Fractional Capacitors
(l/.r)‘~” by a Regular Newton Process
G. E. CARLSON, MEMBER, IEEE, AND C. A. HALIJAH, MEMBER, IEEE
Summary-This paper exhibits a third-order Newton process
for approximating (I/.s)~‘~, the general fractional capacitor, for any
integer R > 1. The approximation is based on predistortion of the
algebraic expressionf(x) = P - a = 0. The resulting approximation
in real variables (resistive networks) has the unique property of
preserving upper and lower approximations to the nth root of the
real number a. Any Newton process which possesses this property
is regular.
The real variable theory of regular Newton processes is presented
because motivation lies in the real variable domain. Realizations of
l/3 and l/4 order fractional capacitor approximations are presented.
F
RACTIONAL operators have been considered long
ago by Riemann and Liouville [I]. Heaviside [2]
noted that the input impedance of an infinite RC
cable is a. Network specialists have directed some
attention to the fundamental approximation problem of
fractional operators. Recent contributions by Lerner [3]
and Pierre [4] employ logarithmic potential methods.
There exists much interest in diffusion problems [5] and
distributed RC networks.
This paper displays a small facet of a non-log-potential
method. Driving point impedances s, so, s-l are to be
augmented by the fractional capacitors (l/s)““. These,
in turn, can be converted into fractional operators by
standard operational amplifier techniques. No loss in
generality occurs by not considering the fractional in-
ductors .?
which can be realized by
RL
networks.
Some work has preceded this paper. Carlson and
Halijak [6], [7], show applications of a Newton process
for approximating the characteristic impedance of a
balanced, symmetric RC lattice. When dealing with
higher-order fractional capacitors, the Newton process
provides approximations, whereas the classical iterative
method based on characteristic impedances does not
even exist.
However, a richness of possibilities [8] occurs which is
frustrating. For instance, there exist n ways of predis-
torting f(x) = x* - a by repeated division with x, and
all yield different approximations to the nth root of a.
Their number
can be reduced to one for each n by ad-
mitting only regular Newton processes.
A regular Newton process preserves upper
and
lower
approximations. For nth root problems these regular
processes are of third order. Such a process is optimal in
Manuscript received May 27, 1963; revised November 1, 1963,
and December lo,, 1963.
C. A. Halijak 1s with the Kansas State University, Manhattan,
Kans.
G. E. Carlson is with Systems Division, Autonetics, a Division
of North A merican Aviation, Inc., Anaheim, Calif.
the sense that the convergence rate at the root is the
largest possible in the set of predistorted functions and
no overshoot or undershoot occurs at the beginning of the
process. The highest convergence rate is attained by the
upper approximation.
PREDISTORTION OF (x" - a) FOR ODD n
The derivation of a Newton process is discussed in
books on numerical analysis [S], [9], and familiarity with
the derivation can be presumed. The notation of Hilde-
brand [9] will be closely followed in ensuing discussions
of convergence an.d order of approximation in the rea
variable (resistive network) domain.
Consider the problem of finding the nth root of the
real number a. The mth predistortion function is defined
to be
L&(x) = (x” - a>/xm,
Olm<n-1.
(1
Choice of m is dictated by a desire for local linearity a
the root, i.e., f; ((~““) = 0. Calculation shows that the
condition is satisfied
only
for
n
= 2m + 1. Only odd roots
can be considered because of a contradiction.
PREDISTORTION OF (I(;" - a) FOR EVEN n
Even-order roots require a more elaborate procedure
for achieving a zero of the second derivative of the pre-
distorted function at the root. Consider the expression
f(x, :A)= y + 1%.
(2
A lengthy calculation and substitution of x = X =
(&I/” =
-
o( yields
f”(a, a) = (n’ - 2qn - n)cT-’
+ (n” - 2mn - n)olnemel.
(3
Setting the coefficients of powers of CY qual to zero yields
nothing worthwhile. However, rewriting yields
f”(a, a) = an-Q-2[(n2 - 2qn - n)
+ (72 - 2mn - n)d-m+‘].
(4
Set q = m - 1 and obtain
f”(a, a) = an-m-12n(n - 2~2).
(5
In turn, the secon.d derivative is zero if n = 2m. The de
sired predistorted function then has the form
f(x, X) = * + x 9.
(6
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1964
Carlson and Halijak: Approximation of Fractional Capacitors
It is a simple observation that the above process and
the Lanczos process [6] yield the same iteration formula.
Traub’s [lo] researches have traced the Lanczos process
to a paper written in 1694 by the astronomer Halley.
It is apparent that
211
CONDITIONS FOR CONVERGENCE AND REGULARITY OF A
NEWTON PROCESS
Hildebrand [9] gives necessary conditions for conver-
gence of any iterative process generated by the equation
2 =
F(x).
The iterative process is then, given zo,
x7&+1
= F&J,
n = 0, 1,2,3, ... .
(7)
Convergence implies that the present “error” is less than
the previous “error”; that is
Ix,+1 -
x,1 < 1% - X,-II.
(8)
But
x,+1 - XVI
= F(x,) - F(xn-1).
Use of Taylor series with remainder yields
(9)
x,+1 - x, = (xn - xn-JWJ, G < .5& c G-1. (10)
The necessary condition for convergence is that
IF’(x) 1 <
1.
The condition for regularity is that
F’(x)
does not change
sign when x belongs to a neighborhood that contains the
root.
The Newton process can now be linked to these results.
It is apparent that the Newton process on
f(x)
= 0
yields
F(x) = x - (f/f’) (11)
F’(x)
= ff”/f’f’.
(12)
One can deduce (since
f
= f” = 0 at the root) that this
choice of predistorted function yields fastest convergence.
Regularity depends on $,’ producing no change in sign.
The minimal condition on $,’ is that $,’ be a perfect
square up to a positive multiplicative constant.
RAPID CONVERGENCE OF THE UPPER APPROXIMATION
A proof is now given that the upper approximation has
a faster convergence rate than the lower approximation.
It suffices to show that
F’(a + h) < F’(a - h )
for a
positive number
h
and (Y = a”“. To expedite the proof
one can assume that
h
is small and higher powers can be
neglected.
For n = 2m + 1, calculation yields
F’(x) = ~
m “+
1
(13)
Appropriate substitution and neglecting powers of
h
yields
F’(cu + h) G
* [yaynh]’ (14)
F’(LY - h) *
e [yr.;hy (15)
y = (2m + lj/(m + 1).
F’(cr + h) A [;f: -$JF’(a - h).
(16)
Therefore the desired result is at hand. This proof suffices
for even nth roots. This is not obvious and the result of
a subsequent section will clarify this statement.
ORDER OF THE REGULAR NEWTON PROCESS
Proof that the approximation of a regular Newton
process is of third order is presented. Consider the error
(Y - x~+~ where (Y = a”“. Then
CY xk+, = a - Xk + @gx; O
= a - Xk f”;,, )f’“‘.
Xk
(17)
Use of the Taylor series
f(a) = f(Xk) + (a - Xk)f’(Xk)
+ (a - Xkj2 l,(xk) + b - xJ3 f”‘(fk)
2 3
(18)
yields
-(a ;,x*J fll(xk) _ (a yc)s fl,,(gJ
a - x:k+1
f’(G)
* (19)
Since
f”(Xk) = f”(a) + (2, - a)f”‘(%)
= (Xk df”‘(%j
the error expression becomes
(20)
a: - xk+l = (a - Xkj3
f”‘(d - f”‘Gk).
6f’h)
(21)
Points & and 7]kbelong to the interval whose end points
are xk and o(.
THE REGULAR NEWTON PROCESS FOR a””
An explicit iteration procedure for even and odd roots
is found in this section. One formula is produced for both
cases but this cannot be surmised at the beginning.
Even-order roots are considered first. The root X can
be considered a constant when constructing the New-
ton process. Then set X = x, since the current x is an
approximation to the root. The predistorted function is
@a
Subjecting this function to a Newton process and then
setting X = x yields
(2m - 1)~‘~ + (2772+ l>a
F(x) = X0 (2m + 1)~‘~ + (2m - lja’
(23)
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212
IEEE TRANSACTIONS ON CIRCUIT THEORY
June
Setting n = 2m yields
F(x) = x.
(n - 1)X” + (n + l)a
(n + 1)~” + (n - lja ’
(24)
The predistorted function for odd roots is
f(x)= “‘“+y a.
Subjecting this function to a Newton process yields
2m+1+ (m + 1)a
F(x) = x*7+ 1)
X
n+1 + ma.
(25)
(26)
Setting n = 2m + 1 yields
F(xj = x. b - 1)~” + (n + lb.
(n + 1)~” + (n - l)a
(27)
Note that (24) and (27) are the same for both even and
odd roots. This is the reason for not discussing both cases
in the section on convergence rate of the upper ap-
proximation.
EXAMPLES OF FRACTIONAL CAPACITOR APPROXIMATIONS
If the real variable “a” is replaced by l/s, the regular
Newton process develops approximations to fractional
capacitors in the form of ratios of polynomials in s.
Networks can be developed which have driving point
impedances approximating these fractional capacitors.
The convergence and rate of convergence of these ap-
proximations of fractional capacitors cannot be deter-
mined in the same manner as the functions of a real
variable. The full problem is not solved here and requires
further investigation.
Networks have been previously constructed for the
approximation of fl when the initial assumption is
x0 = 1, [6]. These approximate fractional capacitors were
used on an analog computer to simulate the operators
fi and 4. The results [6] show that they are good
approximations of the operators fl and 4. These
networks are cascades of balanced symmetric lattices
with unit resistors in the parallel arms and unit capacitors
in
the cross arms. The cascade is terminated in a unit re-‘
sistor. The number of lattices in cascade is
nk = 3nkw1 +
1,
where
no
= 0 and k = 1, 2, 3, . . . , is the number of times
the Lanczos process (equivalent to a regular Newton
process) has been iterated.
Networks will now be determined for fractional ca-
pacitors of orders + and $. Use of a regular Newton process
and an initial assumption x,, = 1 yields
(2%
as the first iterate approximating +a. The second
iterate approximating ‘$/G is
The ladder networks which realize these functions as
driving-point impedances are shown in Fig. 1. Note that
the network resulting from the second iteration has re
sistors, inductors and capacitors.
Approximating networks for ql/s are derived in the
same manner. The initial assumption x0 = 1 gives a
first iterate,
The second iterate is
729~B+15450s~+58375s4+91500s~+69975s2+24090s+2025
22 =
2025sa+24090:r6+69975s4+91500s3+58375s2+15450s+729’
Networks developed are shown in Fig. 2. Again RLC
components appear in the network resulting from the
second iteration.
Appearance of inductors in the second iterates forces
an investigation of physical realizability. Proof that the
real parts of the fractional impedance iterates are positive
on the jw axis is long and intricate. The appendix gives
an outline of lemmas leading to a theorem equivalent to
the positiveness of the real part.
INITIAL ASSUMPTION
r. : 1
FIRST ITERATION
X,‘St2
2s +I
06667
I
Fig. l-Networks with driving-point impedances approximating
+l/s*
INITIAL ASSUMPTION
)(a i I
FIRST ITERATION
3st s
*t=TxG
SECOND ITERATION
s5 + 24s4 + 80s3 + 92s' + 42s + 4
X2 = 4s" + 42.~~ + 922 + 8Oc? + 24s
+ 1’
c2g) Fig. a-Networks with driving point impedances approximating
i/l/,.
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1964
Carlson and Halijak: Approximation of Fractional Capacitors
213
CONCLUSIONS
The real variable (resistive network) theory of a regular
Newton process for the nth root has been presented. Be-
cause this process generates rational functions, it is
natural to ask for the nth root of l/s. Such is called a
fractional capacitor of nth order generated by a regular
Newton process.
The first test in the physical realizability of all regular
fractional capacitor approximants, the test for positiveness
of their real parts on the j axis, has been accomplished.
APPENDIX
OUTLINE OF PROOF FOR POSITIVENESS OF REAL PART OF
ANY FRACTIONAL IMPEDANCE APPROXIMATION
The regular Newton process yields the iteration formula
jw(n - l>[xm-l(jw)ln + (n + 1)
xm(jw) = xm-l(jw) jw(n + l)[x,-,(jw)ln + (n - 1)
= Ameism
(a, + jbJl(aZ + G).
The angular frequency o is restricted to the interval
(0, ~0 . The list of lemmas needed for proof of the positive
real part property is now exhibited,
Lemma
1: If n 1 2, then - (7r/2) I (-r/n) <
(-2n/(n2 - 1) < (-n/an).
Lemma 2:
If
eo = 0
X -1
-
[J 1
, then
-2n/(n”
- 1) < 8, < 0
I
.
n>2
t$ = 0 at w = 0 and 03
Lemma 3:
If
f
I
n22
-2n/(n” -
1) < em-* < 0
then a, > 0
where tan e, =
b,/a,.
The conditions of Lemma 3 are hypothetical and the re-
maining lemmas show that the conditions are real.
Lemma
4: If
n22
-2n/(d -
1) <
em+
5 0 ,
.e,-,
=
0,
at
w =
0 and 03
then
i 8, = 0
-s/2
t w =8,0
5
and CO
I-
Lemma 5: If
i - 2n/(n2 - 1) -c em-, 5 01 f
n22
then 8, > -2n/(n’ - 1).
Lemma 6:
If
‘- Zn/(n” - 1) < em-,5 0
n L 2, and &,-I = 0
, at
w
0 and co
/
1
then -2n/(n2 - 1) < 0, 5 0, and 0, = 0 at w = 0, 03.
Theorem:
If x,, = 1 and n 2 2, then -7r/2 < 8, I 0
for all x,(ju), m > 0.
Proof: By Lemma 2, if x0 = 1, n 2 2, then B0 = 0,
-2n/(n’
- 1) < 8, _< 0 and & = 0 when w = 0 and m .
By Lemma 6, if
-2n/(n’ -
1) < B ,-, I 0, n 2 2,
and e,,,+ = 0 when w = 0 and a, then -2n/(n’ - 1) <
e, 5 0, and 8, = 0, at w = 0, m. Then by induction
-2n/(n”
- 1) < 0, < 0 for all m 2 0. By Lemma 1,
if n 2 2, then -n/2 <
-2n/(n”
- 1). Therefore -r/2 <
8, I 0 for all m 2 0.
A proof that all approximants are positive real functions
in the right half plane has been accomplished similarly
and the lengthy proof is presented in G. E. Carlson’s
dissertation.
REFERENCES
[l] E. Hille, ‘“Functional Analysis and Semi-Groups,” American
Mathematical Society Colloquium P ublications, New York,
N. Y., vol. 31, sect. 21.12, pp. 439-443; 1948.
[2] 0. Heaviside, “Ele ctromag netic Theory,” Dover Publication s,
New York, N. Y., pp. 128-129; 1950.
[3] R. M. Lerner, “The design of a constant-angle or power-law
magnitude impedance,” IEEE
TRANS. ON CIRCUIT THEORY,
vol. CT-lo, pp. 98-107,; March, 1963.
[4] D. A. Pierre, “Transient Analysis and Synthesis of Linear
Control System s Containing Distributed Parameter Elements, ’
Ph.D. Dissertation, Dept. of Electric al Engineering, The, Um-
versity of Wisconsm, Madison, Ch. 7 and 8; 1962. (Available
from University Microfilms Ann Arbor, Michigan.)
[5] J. R. Fagan,
“An Investigation of Nuclear Excursions to
Determine the Self-shutdown Effects in Thermal, Hetero-
geneous, Highly Enriched, Liquid-moderated Reactors,” M .S.
Thesis, Dept. of Nuclear Engineering, Kansas State University,
Manhattan; 1962.
[6] G. E. Carlson, C. A. Halijak,
“Simula tion of the fractional
derivative operator V% and the fractional integral operator
*,‘I Kansas State Univ. Bulletin, vol. 45, pp. l-22; July,
1961.
[7] -, “Approximations of fixed impedances.” IRE
TRANS. ON
CIRCUIT THEORY,
vol. CT-g,, pp. 302-303; Septemb er, 1962.
[8] J. F. Traub, “Comparis on of iterative methods for the calcula-
tion of n-th roots ,” Communications of the Associat ion
for
Computing Machiner y, vol. 4, pp. 143-145; M arch, 1961.
[9] F. B. Hildebrand,
“Introduction to Numerical Analysis,”
McGraw-Hill Book Co., Inc., New York, N. Y., pp. 443450;
1956.
[lo] J. F. Traub, “On a class of iteration formulas and some his-
torical notes,” Communications of the Association of Computing
Machine ry, vol. 4, pp. 276-278; June, 1961.
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