Approximation of Fractional Capacitors

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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



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/,.



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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Approximation of Fractional Capacitors