Solution to Problem 64-14: Resistance of a ladder network

Solution to Problem 64-14: Resistance of a ladder network

Citation for published version (APA):

Bouwkamp, C. J. (1966). Solution to Problem 64-14: Resistance of a ladder network. SIAM Review, 8(1), 111-112. https://doi.org/10.1137/1008019

DOI:

10.1137/1008019

Document status and date: Published: 01/01/1966

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PROBLEMS AND SOLUTIONS 111

Problem 64-14, Resistance

_of

a LadderNetwork, by

WILLIAM D. FRYER (Cornell

Aeronautical Laboratory).

Findthe input resistance R,, to then-sectionladdernetwork as shown in the

figure. Also, determinelim,

R.

Solutionby

C. J. BOUWKAMP

(Technological University, Eindhoven,

Nether-lands).

Divide all the resistances of the n-section ladder network by 2. Then the resulting network has input resistance

R,/2

and the

(n

+

1)-section ladder network is aseries connection ofaunit resistor andanetworkconsisting of two

resistors in parallel, one of value 1 and theother of value

R,/2.

Consequently,

wemust havetherecurrencerelation

2+2R

R1

2, R,+I

2+R.

n 1,2,3,....

Oae method of findingan explicit expression for

R

is to let R, p,/q,, where

p,+ 2pn -[- 2qn qn+l _Pn

+

2q, p 2q 2.

It then follows that p and q are two linearly independent solutions of the recurrence relation

_f+

_4f+

+

2f

0 whose solutions must be of the form

f+

A (2

₊

/)"

+

B(2

/).

Hence,

p

_[(2-t-

/)

+

(2-

V/)],

q

[(2-t-

V/)

-

(2-

/)].

Fiscally,

+

+

(2

R,

(2

₊

(2

and

limn

Rn

%//-.

Also solved by

E. C. BITTNER

(National Aeronautics and

Space

Adminis-tration),

T.

S. ENGLAR, JR. (Research

Institute forAdvanced Studies),

M. Fox

(Sylvania Electronics Systems),

J. M. HoLT

(Collins Radio Corporation),

R.

KELISKY

(IBM Watson

Research

Center),

P.

G.

KIRMSER

(Kansas State

Uni-versity),

B.

K.

LARKIN

(Martin Company),

J. D. Lwso (Oak

Ridge Institute of Nuclear Studies),

W. C. LYNCH (Case

Institute of Technology),.

E. LXN

and G.

STOODLEY (Grumman

Aircraft Engineering Corporation),

I. :NAVOT,

tWO solutions

(Israel

Institute ofTechnology, Haifa,

Israel), R. OnVEC

(Price Waterhouse and Company),

K.

A.

POST (Technological University, Eindhoven,

112 PROBLEMS AND SOLUTIONS

(California Research Corporation),

L.

H. RUSSELL (General

Precision,

Inc.),

P.

A. SCHEINOK (Hahnemann

Medical College),

K. P. SHAMBROOK

(Westing-houseElectric Corporation),

R. SINGLETON

(Middletown, Connecticut),

R. D.

SMALLWOOD (Stanford

University),

S.

SPITAL

(California

State

Polytechnic

College),

S.

WALIGSRSKI

(Institute of Mathematical Machines,

Warsaw,

Po-land), and theproposer.

Editorial

Note.

The recurrence relation for

_R

can also be solved by finding

the nth powerof the matrix

[ ].

Wligorski lso determines lira

_R

for the network in which the elements of the rth section, 2

-

nd 2

-,

gre _{replaced by}

a

-

_{and a}

-’,

_{r 1, 2, n.}

The limit is

R

{+

1-+

(+

1-

a)

+4a}

nd is ewluted bymeans of the periodic continued frctionwhich is obtained

from therecurrencerelation.

Navot refers to

A.

C.

BTEW,

The Theory

_of

Electrical

_Artifical

Lines and

Filters, Chpmn nd Hll, London, 1930, Chp. 3, where ldder networks re

nMyzedin terms ofcontinuntsndto

A. M.

MoGh-VocE, Laddernetworl

analysis using Fibonacci numbers,

IRE Trans.

Circuit Theory, CT-6

(1959).

pp. 321-322, for similar problem.

Problem64-15,

On

aProbability

_of

Overlap, by

MURRAY S. LAMKIN (Ford

Scien-tificLaboratory).

Prove directlyor by animmediate application of atheoremin statistics that

the conjecture in the following abstract from Mathematical Reviews, March, 1964, p. 589, is valid:

"OLESKIEWICZ,

M. The probability that threeindependent phenomenon

_of

equal

duration will overlap. (Polish, Russian and Englishsummaries)

Prace Mat.

64

(1960), 1-7.

The vlue

P3

ofthe probability that3stochastically independentphenomeon

of equalduration

to

which alloccurduring the time

_-F

to

willoverlapis shown by geometrical methods to be equal to

(3tt0

-

2to3)/t

.

The author makes the conjecture that a similar formula holds for n independent

events,

namely

P

ntto

-

n 1

)to

n)/t’.

Solutionby

P. C.

HEMMER

(NorgesTekniskeHcgskole, Trondheim, Norway).

We

note that an overlap has to start simultaneouslywiththeonset of one of the n events. The probability of overlap whenthis one event is assumed to be number 1 gives

1/n

of the desired probability

P.

Denote

the whole time by

(0,

-F

to)

and let event number 1 occur in

(,

to),

where is uniformly

distributed in

(0, t). An

overlap exists if and only if the other n- 1 events

occur attime

.

Independence anduniform distributionguarantee the following probability forthis to happen:

f(/t)

n-if 0

=<

=<t0,

P()