Direct RLC-synthesis of some special cases of higher-degree minimum-resistive impedances

Direct RLC-synthesis of some special cases of higher-degree

minimum-resistive impedances

Citation for published version (APA):

Tirtoprodjo, S. (1966). Direct RLC-synthesis of some special cases of higher-degree minimum-resistive impedances. (Technische Hogeschool Eindhoven : Afdeling der Elektrotechniek : rapport; Vol. ETB 14). Technische Hogeschool Eindhoven.

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

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Afdeling der Elektrotechniek

Sectie Theoretische Elektrotechniek B

"DIRECT RLC - SYNTHESIS

OF SOME SPECIAL CASES OF HIGHER - DEGREE

MINIMUM - RESISTIVE IMPEDANCES"

by

IR. S. TIRTOPRODJO

Intern Rapport ETB 14

december 1966

, '. !!!!!:!!!!'_J!!22r! ___

il_..;!t:._~:._!~r!22r2~.12_

•

.

'

.~'''\'

tt Direct R L C '- synthesis of some special casea of

.) higher'degltee; 'minimum - reaiative impedancea tt •

~.

_I.·

Conaider an ,impedance

z(.)

at-

2n' -th degr.ee t having no polea' or zeroa on the

~. 2n 2n-l

+

+

s _{+~ s}

₊

_•

_•

_•

_•

_{• •}

a

•

_a2'n' I t

211-1

2n

+

b 2n-l a . 1a

+

• •

• • •

•

+

b2n_1

•

+

'b2n

_•

. Let. zeal be minimum - reaistive at n di.tinct . frequencies (J. Cc)

W .

7 II.

t." • ••

JJ ~ •

. ,Write . Z{s)

=

low, aupent the.degree ~ Z(a) bl asup1l1a·· factor (s+q) ;

. '~.' "

r

in which M resp. I denotes the even part resp. the odd part of the po11Dom1ala in

numerator (index 1) and denominator (index 2) •. throu.gh ,1IlU1 Up1linc numerato~ and d~ominator

,A. :. ' .~' ", :" •

" ,

I' .

.Theorem :

If' " to satiafy either

2 / 1 2 , , ' 2 ' t

1

2

(a) ;: s( a

+ "" )(.

+'i.) •••••• (

a:r~) (1) •

or li(a) = , 8 (a2

+

w~)(.2+~")

••••••

(a2+w~)

(2)

a positive q can be found, then the

impedanceZ(a) ia realizable b,direct alntheaia,

with the minillum number (0%: nearll .0), of e1e~ent ••

---,

"

,t' ,

'.

Pro;of I "

•

" "" .. ,

..

I t

(i)

holds" ,we decompose the augmented Z(s) as follows :

Z(s} _, ' __ ' _{Zl s + 2 s} (),' 'Z '{}

₌

· N'( _'~ s) , "

+

"

, '

Mi{ a

+Ni(

s)

•

, 'We point out, that the

high~st-degree

term in

Ni(s)

as well as in Hi(s) i s ' s2n+1 ; therefore, in the numerator of Z2( s) ,'the odd part

(Ni. - NP

is of degree (2n-1) •.

1

,. 'lhe realizability oi Zl ( s) is ~asily recognized by tai ting ' I . 1 ...

!:.2

12

'and applyiD&' the wellknown theorem about the, quotient of

, "ev.n 'and odd, par:..t of a Huni tz-polynomial ' beiD&' 'a Foster-function •

>:;~Wevill, howe~er, resort to an alternative proof for the realizability ofZ1(s) , 'one vhich almost automatically constitutes a proof for the positive-realness of z2(sl •

Cons~der first the real part of the original (not yet augmented) impedance Z(s) ,_

. '~. '

Becaussof its minimum-resistive character at :t'requencies ~,4tJZ'

•••••

~; we mq , ,,:, ',write I

BeZ(a>\ .. · ..

S::jk.>

<:'rh1s

Can

be understood, by recalling, that not only must Re Z(s)

=

0

tor

. '"-,

'ali:!:.1"'"

'8 =t.j~Ao' ., ' •••• , s =:!:j~; but to prevent lie Z(jW) ,intO.

b _./I t1 h ' ddi i al 1 i d Re Z(jCcJ) ,-,'0',

"'0-. ," ecolll.A.l1C nega' ve , t e a t 'on re at on d 4 , } - .&' ...

W~:t1J, , W =t~,

.•.. ,'

W

=:t"l.."

'must also hold, •

,Returning to the" ,expression for Zl (s) in (3) and remembering that (1) applies,

' 2 2 2

(s~wj

've write 'I(s)

₌

s( s

+

w, )(

s

+

uJ,,')

•

.'

•

•

_{• theretC)re} (M

2 +N2), (s+q)

,

,

.

.

( , , ',

= '

(s2..L. L~/Z).2,,(s·2,.A.," T II'V, T ov. '·zt.)2, • • • • , ( 21.)2 ( S'T-~ •

-s ,

2)'

_ N~) (,2 _ s2)

_•

(5)

5=)4)

" ,

I

. I

~ .

. ~. .

f.t)z,

,comPar.1aB···(41

anJ

(5)

,.wesee, that. Be ~l{jW)

=-

2

r

tv'

ReZ(j

wl, •

q . .

.

.

As V8 knoll. that

ae

Z( jW) ~ 0 , (6) means, that ae Zl (j lJ )? 0 too.

<Bat.

troaa.

zeal ..

"Re

zl (s) +'ae z2{s) ; it alao follows • i

·2

. Q )

that : 'ae Z2(j4)

=-

2· . 1 0 Re Z(jlJ ~ 0 0 q=t-r.)

Becal11DC ' tha~ . ~2 has only poles in the lett-hall plane ( incidentally, thisia why ,

ve

insiated on a positive q - value) • (7) suffices: to ensure pos.-realness of Z2(s) •. Due

to

the nature

ot.

the proposed decomposition, ahe numerator of ~2 is of de,ree 2n

while the denominator is stillo! degree (2n+l) • .

, Therefore,the. reciprocal of Z2.· contains a pole at infinity, which m&7 then be removed, leavingaremaindel! pos.-real admittance· 'I,(s) •

Before continuing. let us point out, that condition (1) t combined with the

minilllWl-. resistive property of Zeal at frequencies

"1,

"'1l • ••••• ,

IV,,"-; necessarily

ieads to the additional relation: Mi{s) :rIC

(S2t-~lHs~",'>o

••

{s~,=>

=1

₂

(s).l/s ;.

(a) ..

Utvhich , is a positi"e constant.

. .. .. No., iet us t8ke a closer look: at the pole-removal.

: . ' .. . 12{

s) ..

l/z2(

s) . •

N2

+-.

101

2

(g)

Ii "':"

{Ni -

I

P

•

2n and s{li -

lig>

H2 -

K

,.' Sa:, 1tturnsout, that atter. removing from. 12 a capacitance of value

(11K)

Farad,

,,-L Y ""''' 1IrI.i.tl/ '

the. numeratOl"

at

the. remainder-admittance I"Z becomes an ..!X.!A

Am uea

of degree 2n

2

z

2

r.

J 2 2' . .

" . v . ' L (8 tiVd(s +W,.J • • R • (s two:) . L

" ' , " . ".M:!

+-

(N

I _

N')

' .

(~)

;

in which is a con. stant •

. '1 . 1 2

•

Then, from the reciprocal of Y, .. the imaginal"1-axis pOl&aat.t.1~ • .tj'i, ••• ,:tj",_; can now be removed, and, as Mi(s) is of degree 211; this. will lead to a remainder impeclance of zero' degree ,; i. eo a resistance.

I'

I

'.~ / .'

4

-2 t . -2 t.

. By recalling that

It{

=.

K (a

+

~

)

(s ..,. "'~) •

K

that the resistance-value has to be

L

Ohm •

2 I.

• • (s +~)

;

it· is easily seen,'

Por the case, when condition (2) holds, we hAve only to ehange from

impedance-, ,

basia to. admittance-basi a and then proceeds quite analogously in the manner as deacribed on the preceding pages •

Later on, in the chapter on equivalent realisations, we will show, that this

cbange of baais is not necesS&r7 at all

tor

this ctass of impedances •

' ..

Ve close this chapter with a tew remarks.

Pirstly, the reason why the frequencie8'1,"i" • • • ~~ should be distinct,

is of course, that otherwise our assertion (8) . need not necessarily be true •. '.

'. Por instance, when

Iei=

~ in (1) , then Xi 'to be zero for the same frequencies

as

12

t it does not have to take the form K

12

Is ,because no~ .Ie,)" in '. (8)

can be any real value at a l l .

Secondly, the "f;equencies"

4)=

0 and 4)

=

IX> are exclUded from being one of the

.'. ·.n . distinct frequencies. The reason is clearly seen from the fact that

. ReZ(6)

= .

a2n/'b2n ~,O· respectively Be.

z(c.o)

='

1 •

" .... Thirdly, the' q;" value can never be zero. This is a consequence of the'

second remark, as ve will show now. Z '2.'

. 2 2

z.

2

Assume, that (1) holds, then Hi (s)

=

K (s

*'''1 )(

s +~) • • ~. (8

+"'I..J •

.. ;But by performing the multiplication (Ii,+Nl)(s+q)

=

(N18+q~)+(M18+qNI)

=

.

"

" .k],

i""

Ni j v e see, that for q

=

0 : Hi

=

_{N18 •} < "\.~";~:~<:':',~'~.:

,;', Thus, for q

=

0 , we have:

Ml.

=

s2

(~s2n-2

+. . .

+a

2n_l ) •

This, in turn,means that in (8) one 0'1 the frequencies has to be zero •

As the preceding remark has already rejected this possibility, it tollows that qf: O. When condition (2) holds, an analogous reasoning may, ofcourae, be applied •

' . .

'~ .'

.l!

....

,,,-, :,: . ,'~' , .

.

5

-In this chapter we will take some inte'rmediate steps toward proving an

analogoustbeorem like the preceding one,

ax

namely for an impedance

of odd

degree (2n~1). These steps consist of proving some additional

theorems like' the following one.'

Let the impedance Z(s)

=

be free trom 'any poles or zeros on the imaginary axis and, furthermore

!

be minimwa-resistive a~ n distinct frequencies~.

'i .... ,

""'4 •

. Dote, that in the following ~~~~!!2~_~!_~~=~!~!!2~ will be made ;

in fact, this impedance. of degr.ee (2n+l) is not yet the analogon of the

impedance, of the. preceding chapter and therefore lie uaed the word "intermediate

The o r e m

.

2 ~ 2 ~ 2 •

(s2.f-~

If either, _{11.2 =s (s} +"1)(s+~ • e· • ',.

(s~~~(s~~i.

2 ", .

(s~~

'or _~

_=-

_s

_•

_•

. . then Z( s) is, directly realisable with a minimum number (or nearly so). of elements.

Proot : The lines, along which the theorem is proven, resembles. so much

. .

those of the preceding pages" that a lot of details will be,skipped.

"'Let (12) be the case, then decompose.. Z(s) as follows:

11.2 I) .... (n_l - _11.2) <Z(s)

.zi(s)+i~(s)

=

m 2+n2

+

t. m2

+

11.2

...

• lV '

. '. It can be shown, that lie Zl(j&J) lit

--r-"k

lie. Z(jl.V) , in which k is

W+

.

, , .

".~ positive constant,. thereby insuring positive-realness of ii. Z2

\Atthis. pOint, we should bear in mind, that, owing to the

minimum-. resistive requirements:u;,,::

cl.(s~I.{)(s~w:>

•••

(s~tall

=

clen2/ s •

.. . Removing from the reciprocal of Z2 " its pole at infinity, we have :

...l

JL.. _m ... 2 _+_11. .. _{2 _-_. s_} .• _m..,l/ __ ?n,.-._-_s_(_n

t

· _.

-_n ... 2)_I_c .... 1

y.~=Z\

_{. J , 2 '}

_{c;," .,}

=

_m

~

l - r t1j; - 11.2 " . , \" .' • """'!

,:..

. m_{2 -} s( n_{t -.} 11.₂

)1

c1,

Because . n 2

~

semll cl ,. it 10110ws,. that:

J,

=

m

1

+ (

11.1 - 11.

..

..

. ,

6

-No.te now, that the numerator of

'1,

has'completeiy lost its odd part; indeed, . " it .has. ~ecome an even polynomial of degree 2n.. As the denominator of

1,

is

also of degree 2n .' it foilows, that n j~-axispoles can safel~" be removed from the reciprocal of 13 ' leaving. a zero-degree r~mainder, i.e. a resistance. It (13) is the case, we onl~ have to effectuate a change o·f basis.

In

a w." the impedances just discussed are slightly more "general" than those of chapter I. By this statement we mean, that~ whereas all impedances of the class of chapter I automaticallybeDongs also to. the class of chapter I I t

. the converse is not' :true. In reference [ll,page 185, we encounter the z( .) _ 83

-+

2s2+2s +2 • tor which' Re Z(j) :: 0 and

. impedance s - 3 2 ' 2 ·

s

-+

s

+

s~t n₂ .. s (s

-r

1) ; thus, clearly belongs to. the class we've just discussed. However, on applying Euclid's 'algorithm, it can be shown,that no common factor (s +q) exist in numerator , and denominatorr not even when negative q -·values were allowed.

It could be argued, therefore, that chapters I and I I should have been reversed but in the opinion of the author the chosen order of discussion offers a bit more insight as regards some important details, e.g. the positive-ness

of

q.

Fo.r _{the sake of completeness, we'll prove one more additional theorem •}

2n+ 2n-l _+e 2n_

₁

Let. Z(s) .s. e

₁

s . . .

T.·

0

....

s +e 2n '~-rnl .

-'

_{2n ' . . 2n"':1} :: s

+f

s .

+.

.

_•

.

+f_2n_ 1 s

+

f2n m2

-r

n2 . 1

be an impedance having neither poles nor zeros on the imaginary axis and be

•• iag _{minimum-resistive at n distinct frequencies "','}

~,.

_{• • • •}

_{W""' •}

. ~ 2 !!!..~~

;

l.f. m _ IT (s 2+ lAJ. ) 2 - .11 -t.. Proof : then, of implying

",,=, .

necessity : canonical n_l or 0 and realisability • ;

To- satisfy the requirements of minimum-resistivity,

n1(jW) has to be zero at frequencies:tw,.:t'U'z., ••••

,±WI'k."

But, as nl(jw) is only of degree (2n - 1), the remaining possibility is nl::O.

1'\01- ( £..~

Numera~:i:Re.

_{Z(jtcJ) :: m1(jw)m2(jGP) - nl (jw)n2(j W)}

=

X, \- (".)

+

'1ci~

•

Rememberia,:g (14)- and.~ ~ : 0 " i t follo~s,: that . ~ _{_ m2} •

r. "'h' e' ,

4\

. ; : .

(

2

.!

.

= 0 ' /

.\, complementary case when ml::.H s

+

wi); _{of course implies n2- ,m2} =~~

.of.,=-l

We reco'gnize impedances o~' this class, as being the last "step" . in the

III. . . '".::' ."'""

7

-.' _2n+l s

+

c s2n+ 0

..

s c 1

+

c 2

+

Let Z(s) : ..

n

2n+l s2n+l + d l s2n

+

::

...

.

e·

+

d 2n s

+

d 2n ... l

be an impedance, having no poles or zeros on the imaginary axis and being

minimum-resist.ive at n distinct frequencies ~ r

"i" . . . . ,

w"""" •

. III either being

m_l

+

n_l

m₂ +n₂

the case ,. then

·

t:.:~ =

augment Z(s)

mi

+nJ,

m' +n' 2 2 as follows : • T h e o . r e l l

.

I:C, to satisfy either

•

nJ.

(s) :: cI

(s2+~/~(s2+w:J

(s2+wj (15) , l s • • • or n

₂₍

s)

₌

de 1 s ( s 2+ tcJ,Z) ( s 2

₊

_W*L)

• • •

(s~w

...

~

(16)

a positive q can be found, then Z(s) is

realisable by direct synthesis.

The synthesis-procedure requires one additional "step"" as compared with the impedance-realisation of shapter I.·

Due to the conditions of

Re .z(

s)/

₌

., ~=jw . . .m,m2 2 m 2 2 - n₂ 5

::j-w

Consider the case, when (15) holds, then, of necessity ,

xmxx

xx:xx xxxx _

xf<s~xxxxtb.~xxxxtxxxxxxxxxts~xxxxttE~xxxk;

-

_{2 2 2}

_z

(s

+

W,

)(s +IA~)

•

.

..

( s 2

+

kJ"",H

2. s 2

+

h) o

In the above expressions" k ·and h are positive constants •. No:¥ . we have· to distinguish betwee.I1

..

It" the firstmentioned condition is the case, then we decompose Z(s)

-~ , "

.' _.. , ~:: ' , ;' as follows : z(

8:)

=

,. 8

-.

m' 2

+

m' +nt 2 2 a and, furthermore,

2haS2 is negative fo.r s

=-

jw"

wherea~'

_(k-t-q2).s2 is positive

P"

w::ro.

we have only to prove, that h2

~

kq2 t to ,insure positive-realness of Z2(S) a

The proof is accomplished'l by noting, that c_2n

+

_{la d2n}

+

_l

=

k.

Tr

IAJ,;,'I

"'" y 2 2 i::./

Therefore,.

~·rr~ ~d2ntl

or kq2 _ _d2n+J, 2 _h2

. ""./ ;:::; ... ' I . q .

=

•

1f

W.

. . ",,'" . I '"

" This last statement stems from the fact, that : _•

Now weare in a posi~iont to

of Z2( s) t namely: 1

2{s)

concentrate on the

m'

+

nl

expression for the reciprocal

2

r

..

Note,. that m

₂

in the numerator is of degree 2n

+

2 t whereas the highest

degree. in the denominator is the degree of: .xxx

Pi".

namely 2n+ 1 •.

This. last is caused by the cancellation of the term with degree

2n+

2 in

the expression for (mi - mp •

has a. pole at. infinity, which can be removed, as follows

.§.... . m~ +'.

n? . -

(slci). (mi

m?

+

nj,)

c' .

=

1

_•

then it becomes clear, that

,

which, after close scrutiny, turns outta belong to the class,·described

in. chapter II and for which, except for a conatant.factor, (12) applies.

7, . .

','J:

' 9

-.. "In ease c

2n

+

1

<

d2n+1 we have to change our manner of decomposition, as follows :

=

(c2ntl/d2ntl)·mg. mi - (c?n+l/d?n+l)·ma

+

ni

Z(s)

=

Zl(s)

+

Z2(s) , t I

m?

-t-

_n2 m? +n?

•

lijow;," to insure positive-realness. of Z2( s) by means of considering the expression

for Be Zl (j

tV) ,.

it turns out ,. that we have to prove the· following inequality :

. . . (c2n+1/d2n+l>.(s4+ ?hes2+h2

)1

~

xzixx

1..

s4 - (k +q2).s2

+

kq2}

I

. . . s=;'w S

:jtU

By remembering, that c

2_n '+1

<

d2 _n+ 1 t it is easily seen, that the proof can be . _{2 . 2}

. reduced to proving the identity : kq

=

(c

2n+l/ d2n+ 1). h . (17)

Of course, k and' h are the same as those , defined on page 7 .

On page 8 we've noted, that c2n+l.d2n+l

=

k.

:IT

tc{.Y ••

3cSIi%xri~ltxlx

""'" ""'::0/

2 2 __ h 2• Jf~. Sf . ( ).

,AlSO.: .. d

2ntl-q _'£,:::./ .... • On eliminating the factor-f·orm, we'll obtain 17. Returning to the expression for Z2(s), we see, that in its aumeramor's even part

the constant term has been cancelled out. Therefore, Y2( s)" i. e. the reciprocal

of Z2( s) . , will have a pole in the origin , which can be removed quite easily ..

m'

+

n' ..;

11.-- •

[~

_

(c .

/d

)

m t ..L nl']

d q , 2 2 . c_l' s 1. 2n+l. 2n+l -. 2 I

; . 2 n t 1 · . 1 L _

y _3-,,2· -.y - . - ~-~ -

Y -

2 c s -

• -

---~---~---

s

1

~··"s.ci

1[l4i

1" . ; ¢ : ' ,mi - (c2ni-l/d2n-tl)·ma +nl'~ . ' jpl .

:'~ilthe numerator of Y3(s), its even part ; : given by

-lL

n' h I i ( 2+ wz.) '(:15) we. have : . c· s · 1

= . ·

Ii s ~ " . . . ; 1 .

i-I

-h... t h " m_{2 - ci~·}

Il1.

..

o \rIe also have

"'k- '

2 2 E.

·1iL'

=

(s2+ h ). TI{s2+LJ

.l

G Therefore,

2 . . 1.-

m'

2 CIS • ~

=

s . IT(s +~) •

•

. . ...:::..1 1

-i=!

s. [m;:

+

nJ.

J

.As expected, I

3 is of one lower degree than Z2 • namely 13

=

S' [m2

mi ,m

2

ot' degree 2n.

.

-rn

_{2] ,}

in,. which' nn

1

,.

n" 2 are of. degree 20+1 , and

•

Thus" we have again reduced' the problem to a form (for Y

3) which belongs to the

class, described in chapter II and for which, except. tor a constant factor •

condlition (13) applies

For. the complementaryca~e, when (16) holds, a change of basis is .a11 we need •

TQ,. close this chapter, let us point out, that, for the cases. considered up till

. '" .

now,we have in fact found a kind of continued-fraction development. Therefore,

the realisations itself consist of l~~der-networka, for which the xu=exts elements

are 10S81eS8 networks (always in conjunction with one resistance) ,interspersed

by single reactive elements _•

.

'

I

IV.

.-f: . .10 , .,.

. Analogous to chapter II, which contained a slight generalization of the impedances of chapter I t this chapter will do so too as compared with the preceding chaptereThe abovementioned generalization regards the fact, that although all impedances of chapter

III

belongs to the special class discussed . in this chapter

IV,

the inverse need not be true.

Let the impedance Z(s)

=

2n+2

+

b 2n+l+

s 1" s

or zeros on the imaginary axis and, furthermore, be n distinct frequencies ~ ~ W..(. :1 - - - - 7 WA1,. •

be free from any poles minimum-resistive· at

111

1

+

nl

Write Z(s~

=

. m₂Tn₂ in which ml , nl t m2 and n2 have the usual meaning • NQ;t.e finally, that in the following no augmentation whatsoever will be applied.

Theorem Subject to one additional requirement t if either

2 :t 2 ;:t,.. . 2.%.. n l

=

a1 S (s +WI )(8 +".:t) • • • (s +-~) 2 .t. 2 · . 2 2 . t n 2 • bl s (s +

"0 )(

s

+

W.t) • • • (s

+

wll1.)

then .Z(s)

is

directly realisable by one-ladder synthesis •

---~~-~ 0

---~-ProOif

Let (1S) be the case , then, of necessity

2 Z 2 Z 2 l 2 .

1112 - (s,.

U1 )(

s

+"i) • ..

(s

+

~ )0 s

+

h) _•

furthermore,

. we have .. ; Re

Z(~)

J

s

=

jw

Analogous to the preceding chapter, we again distinguish between and

• Let the tormer holds, then we . proceed as tollows

(21) .

..

11

-Remove f.rom Z{ s) .an impedance

• To insure, that the

. ' (m₁

remainder impedance Z2( s)

=

_a. _---1"--_ ....

m₂

+

_{n 2} not become non-positive-real,

we require, that: Rs Zl (j 41) ~Re Z(jW) or

(s4+ 2h.s2+h2)}

~

(s4+ k

l • s2

-r

k 2»)

s

=

jw . s

=

juJ

That h2

~

k'2 always holds, is readily recognized by using

the numerator of (21) compared with (m

l

m

2 - nl~2) written in

flI,. Jf

k21T W'

,;.;/ -t.

lea4s, :eor the constant term" to :

From (20) compared with the original form of m, we find:

the following . :

its original ~orm,

Finally, from (22) and (23) and because .a_2n+

2 ~ b2n;-2 ' it easily follows ,

that, indeed: h 2

~

k2 •

The remaining condition, namely 2h

9-

kl " however, is not automatically

satis-fied and that is why in the theorem we spoke oit' "one additional requirement".

Again, we p.er1'o,rm the complete development of (m

1 m'2 - nl n2) in its original t'orm.

and.compare it with the numerator

ot

(21) as regards the coefficient of the

term o£ degree (4n+2).· We do the same for the complete development of (m

2.m2)

" ,as compared . with the exp:r;-essiolli for the square ot (20) •

'"" .t

kl

+2f;/,~'

=

a_{2 +b2} -

~bl

We. will then have, respectively

and ': ... -l

2h 1" 2

t, "'{'

=

_{2 b 2}

.From this we may deduce, that .the condition 2 h'7 k l t s . equivalent with

the coefficient-requirement alb

l ~ [a2 - b2

1 .

(26)

For the other. case, namely

a

m ) 2n+2 2 Zl (S.

=

b • m . . . 2n .... 2 2

+

n 2

•

we remove an impedance

In much the same way as in the preceding paragraph, al though a bit more compl icated, we find. that·

,;. thv ·res,t.;.fupction rern~ins positive - real .• ' subject to one additional condition:

,~", .

,

v .. ~ . ' 'f 12 -~

'fhe remainder - impedances have. a zero at infinity resp.at the origin. After inversion, either a pole at infinity ora pole at the origin may then be

removed, to produce an impedance o~ degree (2n+l), which belorgs to those of dh.l!.

Examp;le

A good specimen of·the kind of immittance just discussed,

b'e found in reference L2

J ,

page 4S0 ;' where 'Weinberg proposes to realise

can

s 4

+

63 s 3/S

+

31 s 2/S

+

63

sis

+

i

s4

-r

s3

+

5 s2

+

2 s

+

4

; using Miyatats synthesis. (see Appendix A of this report.)

The above admittance is minimum-conductive at W

=

1 . Furthermore,' (IS) clearly

applies here and aft.er a simple calculation, i t turns out, that (27) also holds.

Therefore" 'I( s) can be'. realised using our direct one-ladder synthesis; as follows

(the complete continued-fraction develolilment is given on the following 'page~

-

13-23

.. Comparing the above configuration with Weinberg's realisation (p.16 of this. report) it is clearly seen, that the main difference lies in the fact, that our method

.,'i

aS

long aspossible" holds onto the one-ladder synthesis, whereas Miyatat s

synthesis from the 'beginning decomposes the given admittance into several ladder-structures, hereby sacrifying much directness of 'approach •

i~ .. ' . ,'" ;' '''i .

~J~'

-~

'--<--J

-I-~

~I~ L---l . / •. .,f -13

.-"

~

~

~ .s... ~ ~ ~ OJ

1

-u

-..w ~

.~

~

,; ~,' "

14.

-,,. .

When neither (lS) nor. (19) applies,' the logical approach is, of course, to try augmentation in an analogous manner as in the preceding chapters •

Al though the theory is not complete as yet,. in particular as regards the various

constraints, without which the one~ladder synthesis would not be succesful ,

we will undertake to realise an impedance, found in reference

1!J,

page 12S

so as to clarify our approach.

Z(s)

₌

. 4

3

2

s

+

lt s

+

4 s

+

2.4 s

+

1,6

s4

+

4 s3

+

11: s2

+

l4s

+

10 is, minimum-resistive for

"'=

2 •

Clearly, neither (IS) nor (19) applies, so we augment with the factor (s+

q) •

/

The value of q which would result in either the even or the odd part of the numerator of the augmented impedance containing a factor (s2.,. 4) , is found,

after elementary algebra, to be q

=

4/9

After an impedance-removal, quite analogous to sll those of the preceding pages, and a pole-removal from the inverse of the remainder-impedance, the result is an impedance of fourth degree.. At first sight, therefore, we've gained nothing" but, as was the case in earlier chapters, this impedance now ,indeed belong to

the class of chapter IV and can be realised forthwith •

The complete realisation will heg be given in the next paper. A sketch of

.the schematic circuit-configuration is outlined below •

z(s)

'..."

A omll R.. L,

e..

(J;J .. 1i; 1

!..t>r

(! L I . I (')'H.J2 !G. L~C t.q) I ' .1 1[.41,.

e

_l

l/..N-e\

Q4U!! Il.

t."

e

0-'(2 1l. L..~C!!: t.~J 1.2)

FQr the sake of comparison, we have included in Appendix B of this report , p.17 the very complicated way, in which GUillemin synthesizes the above impedance.

• (~ ,

.

' , . . . . APPENDIX.

(7)

"

.'

The real part of the given admittance is obtained as

from which it is clear that a pair of zeros ar_e atW= 1: 1_ Since

- 22

the coefficients of the factor of (1 -~ ) are all positive, a possible decomposition is

., (1 (»2)2 (1 _,~)2/,2 4) Re [Y( jW)J

=

-

2 + ... \... + W

=

G

1 (c.» + G2 (w) •

B(e.,) ) B(W2)

The corresponding admittance functions are foundto'be 3 2 1

3s /8+ 3s /8 + 35/8 + ~

4 3 2

s + s + 55 + 2s +

4

We now remove two reactive elements from Y₁ and one formY₂- The remainders are (52

,

_{+ 1 )16} - Y1 III 2 s + s/3 + 1 and

,

1 ~

=

_sZ3 1 + -2 ' s + 1

,.!.2

+ 75/2 + 2 2 +s+1'"5

,-The remainder Y1 ~an be realized directly,whereas the real

' - .

part of Y

2 carl be split into two parts again as follows:

2 2 2 2 2 2 2 2

Re [ Y~ (jW~

=

(1 -

cv )

(~ +

cv )

=

(1 - w ) + (1

-w )

(U

, B1 (w ) B1

(W~

B1 ((;,2)

,

The corresponding decomposition of Y

2 is therefore 15

+ 2

2 + s +

15

•

Each of the admittances can be reali,zed dil~ectly by a continued-' ,fraction expansion.

•

• J.1

I,

/6

-Z IS" 3

-

_3t

·1

3

.3

,

.

I

It

-APPENDIX.

(.B)

As an illustrative example, consider the impedance

Z(s) ==

for which

s4

+

1.5s3

+

4s

2

+

2.4s

+

1.6

s4

+

4s3

+

11s2

+

14 s

+

10

==

" 2

2 4 2

M == m1~

- n1n2

=

(s

+

4) (s

+

s

+ 1) •

We set for ourselves the problem of generating an impedance having

the" even part

•

Theone positive real zero of this polynomial is found by the usual

succesive approximation method to have the value 1.8525. Hence we

. have'

m

+ n ==

s

+

1.8525

o 0

=

(s2

+

0.8525 s

+

1.0793)/5 •

T~e

dominator. of the basic impedance is

5 4 '

=

s

+

5.8525s

+

18.41s

+

+

39.935s

+

18.525

and so one of the component impedances into which the augmented given

impedance is decomposed becomes

(s2

+

4)(~

+

nb)(1~8525)2

Z (s)

=

---:;.-....;;;~----a (m 2 + n2)(m o + n ) o · (a) (c) (e) (f) I" .

=

0.6863(6

2 + 4)(5

2

+

0.8525s

+

1.0793)

(s5

+

5.8525s

4

+

18.41s3

+

34.378s

2

+

35.935s

,+

18.525)

I

.

I

I

-/4'-,"

s +

4 3 2

5(a

+

2&666a

+

6.1935a

+

6.32366 + 3.705)

2 2

(s + 4)(s + 0.8525s + 1.0793)

0 .. 6863

•

The other component impedance is beat found by Miyata's method

. through multiplying Z (a)

t

apart from the factor (1.8525)2

t

by _s2

a

and dividing the denominator polynomial into the numerator

polynomial for a one-term quotient. The remainder ia the deaired

numerator polynomial, and so we find for thia second component

(g)

" Zb(s)

III

a

2

+

20666a

4

+

6,,1935a

3

+

6.32366

2

+

3.2056

s5 + 5.8525s

4

+

18.41a3 + 34.37s

2

+

35.935s + 1.8.525

1 ----~2---2~----~---.

(h)

,Ill

.2

+ (a +4)(6

+

0.,8525a

+

1.0793)

S 6

4

+2.66663

+

6.1935a

2

+

6.3236s +3.705

Aa ,ahould be expected,the component impedance Z has

a

a zero at s

=QD

and the component Zb has a zero at

S III

O. After the

. corresponding polea in,the reciprocal fUnctiona ,are removed, as

indicated, in Eqs..

(g)

and (h), there is left a remainder having

the j-axis pole or zero, reapectively, aa placed in evidence by the

factor (s2

+

4) .. To remove a correaponding, pair of j-axis poles, we

,observe that

+ 2.666s3

+

6 .. 19356

2

+

6 .. 3236s + 3.

₂

₀₂

J

----iJit"ilI"

1 ,,486s

(s2 +4)(s2

+

0.8525s,+ 1.0793)

s2

+

4

In the further development of Za and Zb

t

respectively" this ,term

represents an admittance (series

to

branch) or an impedance (parallel

'Le

branch). The

second~degree remai~der

after subtraction of this

term from the bracket expression in (i)

i~

readily found to be

a

2

+

1.18a

+

0.926

, 2

's + 0.8525s + 1.0793

! I

I

, ,

I ~ , i . . l ~

....

..

i

I

I

. f01\ which we have

as we shouldt since this ~actor ~epresents the remaining zeros , originally assumed for the even part given in Eq. (b).

As in.' the Bott and Duffin procedure t we get two remainder functions. Each of these can be treated by the same method after the minimum. value of its j-axis real part is subtracted.

t .

.... ;1 . '., :; , ~'

..

/

.'

L i t t e r a tu r e _.

.

_'

.1.

W. H. Ch~n 2,. L. Weinberg J.E.A. Guillemin '. >

"Linear Network Design and Synthesis" (McGraw-Hill Book Company, New York, 1964.) "Network Analysis and Synthesis" • (McGraw-Bill)

. ' .

"New methods of. driving-point and transfer impedance synthesis". (Proceedings of the Symp~sium on Modern Network Synthesis t

vol. V, april 1955. Polytechn.lnst.Brooklyn MItI Symp.Series.)

~--~---~- 0

-':' '

. (to be continued.)'

./