SYU'EDY O? fi'fiTHESES 05' RC {‘éET‘e‘s-‘C'RKS,

Thesis for 9329 53995:}; {A ‘55. S.
MECHEGAN STAKE URNERSWY

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6/01 c:/ClRC/DateDue.p65-p. 15

 

Abstract

Synthesis of RC network means designing a network with
only RC elements in such a way that it responds in the given mann-
or to the applied driving force.

But a complete theory on EC network does not exist to-
day and many engineering questions are still unsolved, such as,
complete characterization of all the three network functions ofck
two terminal pair network, synthesis technique using minimum
number of elements,/methods of keeping element values within
practical limit.

The transfer function of a four terminal network is of
principal interest in the synthesis process. From the transfer
function it is possible to find out the driving point admittance
(or impednce) and transfer admittance (or impedance). The driving
point admittance or impedance may be realiged physically in such
a way that the pair of terminals placed at the far end also
realijes the transfer admittance (or impedance).

From the transfer function it is possible to design
parallel ladder structure, Symmetrical lattice, ladder or grounded
two terminal pair networks.

The design procedure of):wo terminal RC network is
quite similar to that of the LC network as were suggested by R.M.
Foster and W. Cauer. So the same methods may be followed to

design an RC network from the prescribed driving point function.

STUDY
OF

SYNTHESIS OF RC NETWORKS.

BY

MAHBUBAR RAHNAN.

A thesis
submitted to the school of Graduate Studies of
Michigan State University of Agriculture
And Applied Science in Partial
Fulfilment of the Requi-
rement for the Degree
of

Master of Science

Department of Electrical Engineering.

1955

'_-- ’«

7,) 32:}.

._<— ('- ‘
,)

ACKNOWLDGEMENT

 

It is a pleasant duty for the author to express
his gratitude to Dr. J. Strelzoff, Professor of
Electrical Engineering at Michigan State University
for his invaluable counsels and guidance throughout

the writing of this thesis.

1.
2.
3.
14..
5.
e.
7.
8.

9.

Introduction.

INDEX

Synthesis of two terminal networks ...

Synthesis of four terminal networks...

Parallel ladder structure
Modified method with fewer elements
Symmetrical Lattice
Single ladder structure

Parallel T RC networks

Bibliography

2‘L.

25’.
45.
52..
63.
71»
84.

Introduction:

 

In network analysis either the network and the applied

impulse are given and we seek the response, or the network and
the response are given and we seek the impulse.

But in network synthesis the reSponse to a certain
driving force is given and the proolem consists in designing a
network having such properties that it responds in the given
manner to the applied driving force.

Network synthesis is steadily assuming increased
scientific interest and technical in ortance especially in
view of increasing needs in the electrical communication field.
The first contribution to the systematic synthesis of networks

wasnade by G. A. Caipbell (ll) in his paper on the "

Physical
theory of the Electric wave filter" in the year l)22. After
one year 0. J. Zobel (73) published another paper on electric
wave filter, which was put into the prOper perspective by R.
M. Foster (24) in his paper "A Reactance theorem" which, we can
truly say touched off a modern n;twork synthesis. Then Cauer.
Darlington (14-15) Erune (9), Weinberg, Gewertz (25) etc, aces
described useful ways of synthesizing LC and LC networks.

But even a few years back, RC networks were playingci

‘ to L3 and

(u
0;

very minor part in the design problem as compar-

go

RLC networks. At present RC networks are lso becoming of

increasing importance as the circuit techniques are extended

to very low frequencies (such as in servo mechanism) where

re difficult to manufacture.

“I

inductar ces of very rlgh quE.Llity
As a corplete theory of the RC networks under con-

siperation doe .s not exist today, we shall restrict ourselves

to the network function which is usually of principal interest--

the transfer function, d.fined as the ratio of the steady state

output voltage to input voltage in the domain of complex fre-

quency variable, P.

sx clusion of inductances from the network in oses

some restrictions upon the circuit trans er function. Assum-

to

ransfer unction has been de-

H
U
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CF
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Ct
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('7
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t o
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fl
(1)
H);
(‘0'

signed to meet some given require enents, the probles enains
of findin3 an RC network which has this transfer function
when ooecated between a soecified 3sncrator and load.
Accordin3 to Fialkow (18), the transfe r function of
an RC network :uy oe r alL ed nhysically in parallel ladder,

symmetrical lattice, ladder or grounded two terminal pair net-

(
...)
P13
”3
(D
"5
L
:3
C“
I
(
9
5
f
U)
0
r—b
(D
C.
O .
D.
:J
(D
C'"
o
’"S
m4
D)
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S.)
1...:
}-Io
N
SJ
é?-
...:-
(J
:3
if
(D
"S
(D

‘ , _ .— /
Orchard (A3- A9), neinoe r3 (63—33), etc.
Here we shall die cuts in detail the uethods sugges-

ted by Guillesin to realize the transfer function in paralle

k

ladder (32) and l.a dde r (EA) networks ard the u3tnod of Orchard
for the development of symmetrical lattice (48).

From the 3.3 c cif ied transfer function Guillenin (32)
derived a network having the form of a set of separate ladder

structures connected in ca Hr llel. In order to r alize the

. "' " -,. ‘C 'h‘ ,3 .fi «,15’ '2, _ ... . -~ 7- ° *- : ~.~. 4-3
network as was coupelled to place tho sore restrictiOns on one
‘. ‘.- .— was w . .. 1 . + ‘— 3 - .- ...fi - -° 3 «.0 -«.' iv ,
proulen. the tirst one out... that one network is Ll ...ininuu
‘. -' ”V 0‘ . " L‘I‘ 4-. "‘ A rv D ‘ f; 1 r. fl “A \ . ,-." f1“~, ‘ (a
phase lunction (i.e. one Ljrvo oi the tr~ ..... -Ler function should

be on the left half of the P - iw plane) and the second one

cc '1 ‘-‘ ‘-.. -.- «_ .‘4 L _—~. -. -.“ -. ~ 1 .3 -—,I~ \ ~" I" . - H - . fl .- - ,1
was tiat the generator should ce 3: zero internal i cecance.

 

But in many filters and equalizers the host important
quantity is the attenuatLon of the circuit and hence the first
restriction is of no consequence. he He ond restriction, that
the network should oe driven by a generator of zero internal

H . 4‘ I“ ‘ . an ‘ ,N, 15: yr I“, rs ‘ . {'1 mu -. . . \A . fl in
resistance m. not us very easy to attain and so it .eeds some

(.3
k”.
3

de3ree of approxiuation.
In the developient of 3.3. network Orchard overcane

the above two re;trictions imposed by duillenin. Though the

first restriction is not very incortant, the ability to depart

F-.

J

‘11.

her

(1L)
0

from this re triction makes the deal n ,rocess more
But to overcome these res-rictions, he had to derive a lattice
network which his the di dvonta e that it is balanced wit:
reapect to earth. But the zeros of this network may be anywhere
of the P plane and the generator and the load have equal resis-
tive impedence.

In his second method, Guillenin (34) developed the RC
network into a single ladder structure, from the given transfer
function by simply shifting the zeros of driving point function
which we shall discuss later in full.

In recent available literatures on RC networks, many

of the authors (5, l}, 37, 33, 47, 57, 61, 64 and 70) dlSL ussed

the desi3n proolen of per rullel T resistance capacitance network,

‘ _

so we 84311 also discuss this proclem at the

(1)

id.

In every synthesis and design proclem, the number
of elements necessary in the network is very important because
that decides the co:t of the network. Fialkow has 3iven a

table (18) of the number of elements -ecessary for different

5—

type of networks to realize tne transfer function:

 

. .P+.§
P zfi -
“(A (2+17 (Pray n+3?
which is given below:

Table I

 

Network Structure No. of
Max K Actual K Elements
a. Parallel ladders:

l. Guillemin . - 3-93 15
2. Fialkow - Crest 10 - -
b. Symmetric lat_ice 0.86 0.86 17
0. Ladder 15 14 11

d. Grounded two

terminal oair 28.4 22 16

Many engineering questions in 3.0. network synthesis
are still unsolved. The most important of them are the com-
plete charact rization of all the three network functions of a
two terminal pair network, synthesis techniques using minimum
number of elements, methods of keeping tne element values

within practical limits, etc.

(0

In the followin3, we will first discu s a~vse the
synthesis of two terminal networks and then extend it to four
terminal networks. After deriving each method of network

synthesis we shall solve one problem to illustrate the discus-

sion.

§ynthesis of two terminal networks:

If a finite numoer of electric circuit elements be

connected in series, parallel or series-parallel forms so that

:3

two terminals are available for exter al connection, the im-
229199? of the combined elenents between the two terminals is
known as the driving point imoedance of the system.

Accordin3 to R.M. Foster (24), it is pose ble to

c+

determine inductance-capaci ance networks, if the drivin3

s hiven. But 3. Cauer (30),

r"
H.

point impedance of the networ‘

(L;

showed that th sane 3eneral method nay also be applied for
dissipative impedance functions when these correspond to net-
works containing only two kinds of elements (resistance-capa-
citance or resistance inductance).

so, we will use the same general method to find the
two terminal resistance-capacitance network, as was used by
Foster to realize L-C networks.

Let us consider the meet and mutual impedences of

4
(D

an arbitrary dimsipati network which may be written in the

most general form by (29)

bik = Rik 4 51k (3w) ‘1 ------------------ (1)

If we denote 3&1- P, it becomes more convenient to

handle.

.1 bik = Rik 4‘51k P’l -------------------- (2)

If the network is driven from the first mesh then
the driving point impedance may :e renresented by

‘71" .m. -------------

'l

where D is the determinant of the network and D11 is the
minor of its first row and first column.

It is clear that both D and D11 are polynomials
in P but due to the fact that there is negative power in the
“element of egn (1), these polynomials will also have nega-
tive powers. So to make it more convenient for manipulation,
we eliminate the n:sative power by leztin3 tre inpedance func-

tion,
63k . Pbik ; RikP 4 Sik ------- (4)

Then denoting the determinate of the elements bik*

by D* and its monor by 311*, tnen we have,

3* : PHD ----- (5)
and Dll* = Pn'l Dll ----- (6)

assuming that there are n number of rows and columns (actually
n independent heshes) in the determinant.
So, our driving point impedance function becomes,

211 : Q : D“ x Pn-l
D11 5“ D11*

or, 211 = D* ------- (7)
PDll*

Here D* and D..* involving only positive powers of
P and D* involves oower ranging from n to zero inclusive and
D11* involves power n-l to zero inclusive.

Hence tne most general driving point impedance func-
t'iOn representing a network in wiich every mesh contains in—

‘kfi3enoeot res stance and capacitance has the form,

 

211 : 2* : AnP14 An- ~1§n l 4 ------------------ AlPino
9211* P (3n-1P~’~¥3n- 2PU’14 -------------- Isislso

: AnPni in-an-li

------ 4..
Bn-lPfi+3n-2'”'i+ -- _______ 4_

These polynonials may be factored in terms of heir

Le: us denote the roots of the numerator and derH01i-

where Po denotes the zero root of one denotinat: r.
The ohysic l realizability Oi/networx denands that
the poles and zeros Cl 3 network should be alternate (30). The

alternation of poles and zeros may be eXpressed by,
H. “.97494?=o -~ ~-m)
—méF’zn-.<fi~—.».4 42- ' °

Which is known as the seoerstion property. This property indi—

cates that the prescribed

N

.eros and poles of the driving point
function must separate each other, otherwise no corresponding
network exists.

form of tne driving point impedance

£41

The factore

function may n”w be written as,

 

7-. \
Z (P): H3(P‘PL) (P‘P3) --------- (r-PEn-li —---(12)
11 P (P'P2) (P-P4) --------- (P-Pgn-2j

where H : in - -------------- (13)

Now, if we put 3w in :1 es of P then equ (l2)

beetmes,

'
‘m -

211 (3w) 3 H (" V1 _
W ('J‘IT- '11:; ) ( 'Ia'r_ “T4 [Y ----------- - (1::- 15211 .. 2

UV

The equs (9) and equ. (14) of the driving point
impedance function may be identified with a onysical network.
But a partial fr otion expansion of this function will repre-

sent the desired network by a series conbination of simpler

So, the gartial fr otion exgansion of equ. (l4)

 

Z11 (JW) : H l 4 Ac 4 32 4 """""" i AQn-E
n ..'- .12 H“..2n-2
----------------- (15)

For the determination of tne coefficients Ak'a
a frequency is taken very near to TX (W3,-W4,...W2n-g).
For this frenuency the particular term containing 3k becomes
very high. So, we can say that when N approaches Wk, all other
terms are negligible in conparison to the ter; containing the
coefficient Ak. At the sane tine, the equ. (14) becomes
extremely large on account of the factor H-Wt in its denotina-

tor, which approaches zero.

50, the equ. (15) may be written when J approaches

(O

Z11 (3W) : H 23, ----------- (16)
w 9 wk 'w -ch

Which is also equal to the egu. (14) so we equate tne equa-
tions (14) and equ. (16)
H AK I H (hr-WI) (xii-Vii) ---------- (W'T

— H2 3—— 11 ----- ( l 7 )
‘W'- TJ}: W2}: - vii-2 ) T‘H.’ '54 y ---------- ( Vii-'1: 2 n — 2 7

A}: = ( ’1':- 19:1 ) ( V’lY-Yn'f ) ............. ( 23".va2 n - 1 ) ------ ( l '3 )
v v ' j 'I.|‘r_'..'f1{ _ 1 ) (If- YI'VTIIC ‘i "' " < {I - “‘2 n .. 2 >

Now, to realize the ne work from the partial fraction

eXpansion, let us consider the diagras of a parallel resistance

capacitance circuit given in fig. (1)

 

 

 

 

Its driving point imoeoance is given by,

211 : 1 ------------ (19)
l
RkI JWCk
- RK
1 l JWchk --------- (20)

But at the resonant frequ;ncy,

Rk : - . 1 ----- ('21)

 

OT, Pk

ll
L4.
C
r?‘
I
I
H
I
I
I
I
A
ID
ID
V

 

So, equ. (20) may be written as,

2111: {1.5. = R17..."k
TI 13 wT-v """" (23)
'jwk . it; I

 

*3
h o
...)
(D
:5
$3..
"1
w
...—I.
:5

l ( x}
C
c ‘1'
Cf.
i...)-
:3
d
' D
<1
9
C
(D
0

if Wk in the numerator

 

II
I-J

‘,,' 7f
H " Wk

Equ. (24) is conneroble with individual terms of egu. (15)
excepting the first two terns. do, tgrns of equ. (lj) may be

represented as a simple oar ilel resistance-ca acitance network

’0

and when all of these two eleneit network: are connected in
series, we have tfie driving point ihpedance as éiven in equ.
(l6), excluding the first two tarts.
we can find out the values of resistances and capa-
citances by the following way,
1.. a 211 (w-vk) ----(25)
Ck : Y11

. v
.

------- (26)
H. l‘:{

there Y11= z11 ‘1 -----(27)

L

Corresoonding resistance nay be found out from eou. (22)

Among the first two terns of the equ. (15) the first
one is a constant and may oe represented by e resistance. The

second term so may be represented by a capacitance
w

R2 = H = ‘33- ------- (29)
n Dn-l
and Cc: -AO ----------- (33)

Hence the physically realized network for the above
driving point iooeda ce function nay be given by fig (2).

Now, we will solve a numerical problem by the above

Let the poles and verse of the driving point function

’4: 8 ------ (31)

-So, we can write the driving point injedence as,

2 ----- (32)

Z = (P) = H (P4 '5
11 P13)

2) (PI
P (P¥“) (

} .
= H PcIBPI 12 ._.
Pi412g¢32P " C53)

Now if we put pajw, the equation (35) becones,

ij : H (-w24121 +‘83w
-12w2-J(w3-32w ------- (3')

by rationalizi 3,

211 (jw) = H 4w44112w2~1Lw5-52w3-384sl
W6430w4¥1o24u£' ----- (35)

ll

Let us accuse the real

Vic—)0

zll(jw)- H l

i...’
I\) A}
II
H

H
c)

I H
C.

O

i:-

 

112 : 9.13

 

C) C)

ox 4>

II II
‘5 :3 to
' E-J

r—I
hl
F"
(D
*‘S
(h
I]
f...»
U‘
(~1-
L5
i.)
l

K 1
'r

1’)
D
O
(J
,4.
CI‘
5.);
z“
("I-

gart of 211: 1, when

.,-4.-, -.-,” , . ‘
.paoio noes are cilcu-

Here R2n:o because 211 (P) vanisies when PE &C

71‘. .- 1, " , :
ine networ; 3 smn in

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9‘1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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FIG-LI-

 

 

15

same manner as we have Mir ady done by exooncing the driving
point inoedtnce function into partial fraction and comparing
the terms with siiple coo:oonent networks.

Here Yll: 1
le

malfflIfimzinflTI """"" 433
AfiPfiIKn=lPflftI -------- -¥so ----(39)

P (Ba-leV-‘msw 2-2.: “Iii ------- +5; ----(40)
A13?“ ”It; -------------- no

This may be expanded int: partial fraction eXpar sion.

. y11- P é; I A: ------- I AEn-l ,,. Ln.)
P-Pl P-Po I P-een-l

when P app preaches 'k (by the same argument given
to derive equ. (lo) ), the admittance function Ell nay be

written as,

Yll : P gr ----(42)
P

II
[.4
n
.U
I‘d
P?
“f
I
“P
*0
W
I
I
I
I
I
I
I
n-
\JJ

AK

 

Now let us consider a series RC circuit as in Fig. 5

Figure 5

It‘s driving point admittance nay oe written as,

Yll : zii?i : 1 ------- (44)

: ch
lljwska ------------(45)
By equ. (22)

Y c lick =P3k X Pk ----- (46)
11 jw7ejwk Pk-P

Yll : PCk x l/-Rka = PZRK ----(47)
Pk-P P-Pk

(41). So a parallel COflbinitiDU of these networks will give

equ. (41). The resistances and capacitances may be found by

HR : P 211 --------- (48)
P-PK

and Ck : - l

in figure 4.
A num.ric l exanple is iiven celon:

From the last exazple,

I6

Yll (P) = .11? (P+ %3 (Pl8i --------- (50)
(PIET (P¥é)

 

 

Iii (P) : A23 4 ASP 4 nlCP ----- (5-)
§¥2 £10

From equ. (47),

 

 

 

 

A2 = (p422 x .llP (P+4) (P}3) ’
P (P125 (PIS) P : -2
: .11 x 2 x 6 : .33 ------- (53)
4
A6 3 (P 4‘62 x .llP (PI 4) (P+§) P: -6
P 9+2 (PI6)
= 011 X -2X2 2 .11 ““““ (54)
-4
By inspection it can be said that,
A10 = .11 ------ (53)
From equ. (43)
R2 : _ 3 l = 2 23 ohms. ----- (55)
:12 44
R6 = l I l 3 9.1 ohms. ------ (37)
A6 .11

1 =1
,ien; 2x2.3 = .18 ------(38)

O‘\| H

X 9-1 : .0135 """ (59)
and 310 = A10 =.11 --------(6o)
The network is sgow; in fi*ure (6)

Jiuer s extension to Jester s reactance tneorem (30)
may ilso be applied to the synthesis of two terminal 3.3.
t

works. so, we may get two cannonic fOfis of ciysically

D . 1, ;. n . .;- A. -1 . . l r ... V. f 1.. ‘ ‘ ... ,
Ii he cohsi er » Inuucf stricture es snotn in fibure

0.81103

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1.11

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u
1..»
C"
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i
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211 of this network hev ce expressed by elternste seri and

._\o "1‘. , ,0 AW... ,. Spa ", 2’9." .. 1. .- "
Acin-tions 3. tne -csponenc: of the “:tuwrz, starting

.' 4" 1: If" F» ,"I .”‘.\."‘-‘ .- «5“ -.. _'-2 a ‘ "o" *1 ‘ '1 .44 "H ‘ t’ ' "g n :1
Mina the TL- LU -. wilt/L '15:..-4 ii',. ‘. 7‘?" AL)”; Dinah. ta)..«;«1"u («ILA-4 II-r.'.i-'-u-~ S

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u
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’3
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k.
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(D
3
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U
cf
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'15
r I
o

m = Emmi-W“ --------- {Am 0 : 2.
Bn-IPHTBn-i‘l’fir-H ....... 49,9130? 4

t8

 

2-28

 

 

 

 

 

 

 

.meg
T l
Fifi—6 ZN.
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F16 — 7 R |
———4AM/\v— __4/WI -———-MAm-—
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FL r ch-
fi' ____/th___
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C1 11%

 

 

 

 

 

 

Fifi-8

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le : El

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= a: I l --------- ---(61a)
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so, it is _;-r .u t i- :e .3 o

t, we o'n drew a licder
3:re:31te£ by RE', and 348 cap
LJC .3; 3:': in tnxa'eiu. (62).
inre f:r:.agive;1 in tilirse (3)

fi_f ‘3 *Y“\
‘KJ‘AL‘ .2

numeric l e le

1

:11;

networi

1 7 .3
LJL‘r—J

-ueu frsct
r'2 W” ‘7 A" "‘
“1:13 U v-Lj «:1 J

of wn 3n

. O v
n w-J

J— .~ -
. x J \(‘\ I 'r "’1
u&;.u \.,A LAD.“

A
V

be

20

I

1|

 

Let 211 = (PIl) (PIE) (PI?) --------- (:3)
P (Ilei (Pia)

 

: 33$? 2423P+1§ ------------ - (64)
P3¥3E24 3“

\filr—‘h-A
’U
4-.

muH

CVHHJ
’U
...—

rflH

So the network will be of the form siown in figure (9)
Another fart d? cannonic structure new :3 reeli7ed
if we turn end for e:i both tne nuner tor a“; denoninitor L
;rocess of division and inv rsion is
carried out, a centinued Erection will result in Wnich alter-
nate terns involve P-l. Her:, the iniedence function is,

lea ApiAlPI ----------- ~+Aggn = z ----(65)
Beware-TI ---------- $3an _ y

; 01’1 P’1 +§1 - 31‘1 P‘1 *i ------ (67)
y .

where Cl : A and E; is the eiainCer.
50 X
The ease process of division end inversion is carried

where R: T1

7’—
s1

. r 7‘1 1 2 0. R-
and XI : L2;P 7-‘-“--' ''''' {oann ------ (63)

21 AllP*---------*----%An3n

" —‘ . .r .... ‘,. . ‘\ V ,. -— - v 7-,-'..
in thlS may Wufwlfl :01 in cycles, he UgVfi,

a

211 = c1‘1p‘1Il
Rl-l I;
32‘1P‘1I

luHH
[U
I
...—J
‘4‘
(NH
:3
I
H
*d
I
H
....
xflH

1/1”].

P
r

*iven in fiSure (10)

a.)

The ne two r}:

t *J
U
"5
(1‘
.1
L
O
/'\
Q
J
V
H
(

A nunerical exmiple for tne eoove procedure is Shown

" '»- - ur.‘ 4 "
i‘rom tile last; 82.1.4115”,

I P: - ------------ (70)

 

- - -1 Hi - 2
z (.33) 1 P 1 I 11.7-iiz.123.+9«
33% 03— I P3

Continuing the sane grocess, we have at last

211 = (.53)‘1 I 1

ine network f_r this function issmown in fig. ll.

23

I2

_..IL-—— 1‘
or" 30

 

 

 

 

 

 

 

Ch
J1

 

FIG—q

4:

 

 

Pia-40

-005/

.98

, J1 _
H 1F 1r
[47% 3-3 % 1 50%

 

 

FIG-u

24

D I” . '71.. v » j ‘1’, y 1
i roar reunin’ai feta-bras:

Syn'hesis o

A four terminal network may oe .escri ed as a group

J

(D

d . ... , A 4" ... r - . t .- . ~ , , r- . fl .- - r‘
o: inceoahce el neo-s (or eieuent) having one pair 0; teruincls

 

for connection to a source of oower or any other supply network

(which is known as the input terminals) and a second pair of

L

tereinals for coineotion to a load or any other networks to
which current is delivered. This pair of terminals xv: known

as the output terminals.

J

ter inal networks (which contain only

‘ - 4‘"
T118433 .I."

k)
C.
r5

{-21 .I

resistances an cspsci ances in this particular case) may be
physically realizable from the oecified transfer function if

35)

A

this transfer function ooeys certain restrictions
. “v . -_ .
Tnere arelfew different procedures to develop the
four terninal R.C. networks from the given transfer function

and three of them are given below:

2f?

1. Psrallel Co nbin tio n of Ladder Structure;:

 

Let us consider the four terminal network shown

 

in figure (12) and inve sti5-.te toe Characters of its short
circuit drivin: oolnt .zc transfer -dsittdrce.
I, IL
__4___

ET _ N IE.

 

 

 

Figure 12

The relations between the terminal volts es and cur-

(D

rent of a f or teriinal network hay jiven by,
2

H
H
II
r3
H
[‘1
1.4
4-.
H
1.4
IL)
I.

- ---------- (71)

222

H
4-.
I<
m

12 : Ylai
in which ill and :22 are the short circuit driving point and
112 : :21 (the retwork is a s\med to be passive) is the short
circuit transfer adtittances.

If the network K contain; only resist noes

and capa-
citances, th inl and 122 are ational functions of cosplex

freouency variable with sioole poles and zeros lying alternately
on tle negative real axis of P plane and the Stalles: critical
frequency is sero (33).

find out the charscter of Ila we reduce he four

tarhihal network into a two teroihal network by usin5 two

. ..n c.- - u- ; w - :. (a:
trans o h<rs as shown in fi5.(i,)

 

IL I‘ I“. [L
1

ENE N 35 t Tre

 

 

 

\F;

 

 

T;

 

 

 

 

 

If the transformers are ideal tnei by eJu. (73)

I1

agilisi i abyiese (72$

12 - aby1231+b§Y2222 (72b)

where s and b are tirf ratios of transforsers fron fi5 (13)

U i
O
U
a;
$1)
04
Q.
P-
:3
t l
(a
L,
A
N
[\D
} i
V
(o
04
A
\1
l U
0
1-1
(T)
is
<',
(u
U

I. aeriie 4 Bab y12 s 4 b§Y223 ----- (72e)

or we cen write

to Brune (9) and 3ewertz (2;) y has the
same orooerty as Ill or £22 for ell real values of a and b and
tnerefcre ylE cen gave only single pole: restricted to the
negative r;el axis out tners is no restriction upon the zeros
of y12 which may occur at any slugs.

=‘eunent

2
d
(D
:5
U“

t‘:
(‘+
D
i
(U
L)
((0
(D
So

0 .

' v v ‘ A .- - ' ‘ 1. ’
ROW, Wucn.PcM%I©zCfl;S :5 ' \

used to derive eau. (16) ) we have

,\.I

Y : K Yll - Kll‘
P-Pk P-fifi

P?Pk PePk

Kil Y22 - K22

PaPi -Pk P-Pk e-ex

27

here K, Kll, Klw e.d K22 ore residues at tne pole P:Pk so
23° (72) bec.nes,

K : K115? i K12 so f K22 b2‘ ------- (73)

e tniorea, we IQOW that,

01-)
'"5
(J
L1
C)
Ru
C
(D
*‘5
U)
’1
(D
0)
r4
Cx
CZ

K112; 0 K22 ;; C) -—-—(74)

and K11 K22 - K112 2? o ------- (75)

which is the desired c1nditicn for the @fiySiCAl reelizetion
of a four tallin l R.o. network.

H
O
to

etiefy the sou. (7;), if St in; point K12>O
then K11 and K22 must be drester than zero. so it is clear

that if Kll or K22 has a pole at some value of P, K12 may or

to

may not have the pole but i1 K12 has pole at any value of

P, K11 and K22 must hove that pole. This may be expressed in

AAA

another wsv that if the de5ree of numerator of yl2 exceeds

that of denominator (by one) then the numerator of both yll
end y22 are one de5ree ni5ner than tn;t of their denominators.
‘3

Row, let us consider the trsnefer lunction of a

four terninol network with a lood resistoroe of one ohm.

 

1:
—">'_1 ‘

f E' N E21 ‘ UM

 

 

 

 

 

The rins?er Function of tnis meteor: may 'e written

SD
(’1

£28

 

. m e .1... h
s \/ ...- .H r h r C
.l A. G T 0 1o 9 1L .1
1 .l a 1 . “a . o d .. e
2 (K 1 2 .1. 1L .1. U u 1.: W n
1 r 1 r a .. 1 .c .L t
, .31 e d ..1. v1 1 o u.“ 1 c 2 n
”I. fil.“ ,wv M... V 3U .NHH o...|_ p L
o1 u .1 11 f 1.1 8 2 .3 .l O
. O ....1 3 O 9 .1. 3 u .1 ..o
.1 8 fl 1.; I 1 I .-.1 -1 o
mi mi 1 3 no 1...: 3 1.: . 1 ...
o w .l «1.4 9v n, L f“ L e :1; 1. .n
t .1... 11 r .1. .1 1 o 1 .1 ,1
.4 .1 . 1 ..-u 3 t. 3. C v 1 3
H d T. v S y _ . _ fl 1
.l ... 1 C D. n . 1 and
.2.“ 1.1.1.. r G 11 . .... l 1
. 1 O 3 r . e 1. 1 n 1 C .1 1 3
fl . . G ...h L C 3 .1 ..
3v .U +U D . .J \U «1“
....1 3 )1 .. . .l T. C 3-1 . H a; 3
U a .. l 1» ul \/ o \J, 1;
a, (-1 1. r... 1 o. 9 A o C
..u 3 a L ... 5 / ..h . to v...
C . 01 :1. .n1 _ .1 . u 3 f\ 1). ML. ..-.1 3
1 o 3 a . H L ..H .l 3
n 3 1... t .5 S ..J .11 L
i .1 1 .13 u 3— .1 u x 1 . -1 .1. +U AV
3 .1 1t 1 .. 1 t t .. J 1.... 3 u
2 .1 _ VI— H.. ... 1. 5,. ha ..J
2 11.1 a oi T ....n C 1.. 1 1 . . Z
\/ ”iv fi_ a. O 91 L c Fl“ ~_ 1 «u .D
6 \/ -.. 1. . C . d e 3
7. 7. ..T . r e .1. . 1.1. 1 1.1 n ..C.
/\ 7 1 t O .l .1. ..L ,3 1 o E 1. 1c
. (x t X C ...1 P T ....1 n1
. . VJ .1 0. ...u w. .3 .1 o a ..o O J J
. 0.. _ 1 o .1 31 no .1 1..” 3 a. 1..” .1 .L
_ 1.4. . .1 .rl_ 1 .1 H . 3 .1 1 w; 1.. .x 1.
. o1 . 1., , ... O . . 3 u .l .1 T
. 2 2 O... _ u 1. 1 c 3 .1 . -1 a 1. 3 t 3
_ d1 9. Va _ _ .1.u 1 T s“ 1 T .. u 3 ...u.
v1 v1 _ .... n t . .1. e e a 1 o .1 1 u
l 3. j. .1 f 3 mi 1.. H. ..-.“
2—1 .....3 21 1 2 C o Y t a 3, n. s. -
T1 ... 3 .... v1 9. e .3. L _ . 3 .l ..o H . ..1
1 D; H h S t ..1. 2 )1 T _., .. . 3
: E .... . 1H... .1 .. 1 O t 2 1.1.1 ....“ .... . t
v, 1 l .1... 1 1 V1 ",1 +1. 3 a 1 S e
2 1 D1 0. 2 .l 3 1.; \J .11 n .l 3
3 E 1 1 1 : u. 71 .1 7 f H... ..." ...,. 1 u 1. r
_ 1 Y y .1 7. o o r t e .
2 F ..., e /\ S .3 P P
= : : : 1 d h .1. S 3 n t ._
Va t +o . e e .l t 3
2 e ..u 1 1 r 1 ....1. a, e
W... F ..u D n G1 O C e h . u n n
3 1L :9 .l e D. U. .3 t .1 c 1 1L

‘-Q

"7

-A

.311

24,1. 1+:
akAAQJU-L

{I

LLQ- ‘4.

LV'

Wt“

00,

\ 1"
JAA

‘.
I

m

1‘

.4
vV

Y12 : A {P = Anp?+.An-1Pn-t4 ------- { Ao

5‘5, snpn%3n.1ehrll --------- 1 3° ”“‘(73)

Afl‘ " " "1 "‘ I? ‘ r '\ .
gfleifdJMLQ .1? ,VlJA m1fil 1 (;) say .0 uritten as,

I“
H
[U

 

O
Q
Li
'0
\0
"S
9.1
D
(J
S“
O
/’\
( l.)
H
V
H
(+-
I
(L
g 2
,C
O
.4
«J
/

1&9 p lynvulzls :(P) should ;3 gartea 1n SUCQ a may
~.,. ., «u z o,, , .- .. . ...w-flrn. .
tuna tge r:Llo : (P) 3 (P) 1. thblcillj F€a1lxa013 r1.1nq

ooint a u1t1&DC3 of an R. C. nctwork. 319 :revious
sacwei th?c to renliye the nengrk, y22 snoulj got start witn
shunt C"Uaci;in;a but he a pnjsic‘1 r31117acillty aiso‘denanas
that y22 snsuld not have any 9313 it P=D. This reszfiction

can be 8 ?:1y get by 5332*2;inj 3(3) in sucu a may tuit 5'(P)

and B" (P) both have tge 3113 Ja

iiure:

F PLANE

 

 

 

27

'30

H
{"5
1
O
D
(.f.
,1-
.-J
(D
\J
(‘7'
R J.
4
IL)
:4
H.
(I)
I
)
5‘
.
0‘
F1.
;.
(J
(l)
\)
II
1
$7
:2“
C;
[I
.
H
' i
5‘
(D

)u
("4
k A
H
L\
(+—
(L
<.
r-

C
l

C)
I‘!
LL)
5‘
:3
Q.
l 1
9
C1”

f r ? bv

Q)
’1
$0
5‘
C'“
L)
(I)
(3
Q
<:
(D
U1
0

3 polynomial E

H)
H.
r

p
{)1
( f-
I

M ‘
( i
C1-
'-

V
(L.
N
CD
"S
O
(D
O
H.)
cf
L
(‘1

each point, we will

lie in between the 7ercs of the polyhcmii s 3' (P) and B" (P)

(D

... ‘ H ' ‘ . f.) V- ‘ .1 ' I 1‘ ‘ K - ’1 -
lue exi. (,_) he] now he ez:.nue to

112 = A0 Al? 4 --------+AnPn -----(84)
"".._T/ .. 7‘ _ '
3 .P) 4 a :) JIES
It is a well known fact in the field of network sy -
thesis that the drivin5 paint edgittah:e functica Y22 may be

e vurietiys of ladder sprac;u*es with resis-

and it may shown that differeht tygesof leade: HGtEDFKS Which
have the driving point fidmittahee Y2? hay be game such that
their shjrt CiPCJit trihsier Litittuncas can as excressea by
any one of the separate CSPAS of equ. {84) egg frag this we

‘C: r n -~ 007 a "‘ t ~, m 4:} ,2. a 3 C .vA l,’ ““9 ... 1—2:; 5‘ .-\ "o t‘v‘. 74,»: A”

.J Vu --JLU.» ~— ..,-p.,t - Lie SJLA ullvuLd L Jr. . 41“"J by ugh“, Li- Jéil a

.-‘ . ‘ t . .
I" ‘ -1-~. ~‘ -‘ ‘ 53v. ,_ :2 ‘n' ‘rr n! " ‘~ 1 . --. 1 '- 4‘ ‘ '
Get 0“. SKAK’L; .v-JULI‘. ilk: Uhv- .L.. ‘IILALLL‘L ~41 T: 1J3._:4

(D
( 3
cf
(L
t J.
:3
’0'
ti
"S
gr.
H
H
(D
H

and irfieifiuglly gultirligg 0y 4 seile factor so as to ootain

a esultart hezwzrk wit; the i jittgnje lid.

t
m
U)
(D
D
O
:5
cf-
20
S.)
(.2
0
<1
tD
S Q
b o
L.
a).
&
U}
(D
O
:3
(D
C"
S”!
"5
C+
C’-
D"
(D
U)
k:
D’
I

$12 = As; ALP} - ----- - { An?“ (33)
an TP¥leTP*42)--IP¥d£n-l)

9|

whereo («\{qa ...4qznfl‘4oe (86)

H!
D‘
(D

polynomial B'(P) will have the form
B’(p)= KI<P+CXL)(F+QLP> (moan) ---(87)
The valuesof(ik's in the egu. (87) Should be so chosen that,
OLog‘<o(L<o(_b "'4dzn—‘40‘Ln4’0 (fig)
and the value of K' is such hat,
Klaldq 0(1)“ 480:5: gna,o\5-~ dun—I "‘89)

We may now foriithe oolynoeial B"(P) by using equ. (79)

LU
, I
’U
V
II
U)
*U
V
I
LU
.;
V
I

I

I

I

I
\L)
O
V

. a” (PIag) (2+as> ---(P4dz) ----- (91)

where, OAQmKdiéaz "'édI—x-L 40(2 Adah-yéazg‘ ....Q‘IZ)

222 = 3"IP - Ehgflkgn-Lgfl?t Eo ------- (93)
FTP DnPW'DnelPi‘g‘.‘ D0

 

or we can write

 

Z32: Du]??? m‘IRan‘I’ """" +22 = DC?) ---(3 )
EnPr¥?En-r95ut¥’ ------ {Eb 3(2)

Dividing the gen mihetor by numerator,

222 : Hi I D'(P ------- (95)

4

--1
.4

where R1 3 23 and D'fez is the remainder given by,
in an?

 

D'(P) : D'n-lpfir‘I D'n~2phde4 —-+ 2'0 ----- (36)
E???“ EEPEIEnllP9={I_------¢I Eo

By inverting the remainder,

clPI e'ng ------ (93)
D \P/
fiTn-l

and E' : E‘n-i-£fl='HE'n-£ZPF‘=’3+ ---+Eo ----- (93")
37' .D'n-flPfi-L¥an22PDe3FF---IDo

211: R1 I

cycles 0:
we obtain,

222 : R1 4

(16) where

l
c

The quotient of the polynomial V’ is treated in the
same way and after another cycle we heve,

.Jt

. n ., .~ ' o '1 ‘2 ,— ,, at " ‘
n the Safle meaner D' is cealt eJHin use after n
‘11’ .

I
41;

shtinued frectioh the remsinoer is e cshstant 3Ld

paw

J

P I

If 1

’-

37 I f—J
:3

...-
H

f‘l

he fore of the hetwo k realized is shown in iigure

Rk's and Ck's of equ. (101} are the resistances

32

33

of the network.

This network no

(.0

the scort circuit driving point

'1

adnittgno Y22 and the snort circuit tregsier edtittence

Y12 -

:1
O

: (P) where so is a positive real constant which can

L:

be found out by writin? the 4;:wort efinittence in the matrix

A, n I‘ p. . ,. 1‘ « ms pn‘ .- r .2 . .0- '~ ". 7 ‘. '
form 34;; 11::er tug fJfifl.;L\llfJI 0 3‘1” :1..nlbt»;fi‘39 ‘-.~.;icn is

given oyi

Y12 = 912 ------- (1o2)

where D is tne determinant of the netwirk adtittence and D12

is the ninor of its first row and second column.

onsioer the function £12 again, but
mmL

both numerator :;o oenominetor/turreo.en; for ego.

‘v’. y L
i‘zejht, 1.9 U U.

(I)
0

Do and the remainder

E' P - 3'1P&3523 4 ------ IE'nPH — ----- (105)
D,P 33¥D1P I ---------- IDnEfi

 

Now, we divide D(P) by E'(P) So the equ. (104) becomes

where tne reneinder is

on
J.

------+TN.')

2 4

----—---(137)

 

 

.I~

.lu

&
3
Wu.
«.3

0L

3
e

3

'3 *1

QPGVlou‘

HQ
-éJ

to

lSCiUCeS

.1:
JD

rt

8 P

t to

ind ou

p
L

to

\
'v

a
1.

Cu

0

M8

0
J

\I/

20

no

8
3

n
.fL

:1.
AV

1

L

be concere

u

Lfl

.rk.

~ ~

‘I .“

*8
uvfi

' 1

(‘3

_ . c
oreri

C)
u

.h t;

.2 O-
I I
.A- v

W

O
‘

in;

x..-

\I.’
Go
0v 1.
..l. ..T
( A
_ 1:3
_ 3T
_ “I
. fl
. lro
_
_ I
_ n
_ a1. _ DD.
_ .
. .
. .
.
1
If
P
.w . _
wise
I..-
L
1:3
If.
1
_
D.-
1
1?»
IT
1
n3
. _
9.
9.

.1

('73
.Av

“J t

.1

ea

7-.

revli

1v

’1’?) .

\

figure

. A
iven in

r.

(
\

3

"3

no.
n

.I‘
(...;

n; A

a
L t.

)

f“
__

(13»

sea.

1
"‘1
w

,1" 3 ,.
.V‘l'J.A

. .4

vv‘ .
a
I.“

also

35'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Rn+l fin 92. R,
W—Wr —- —-— M/WVL———-/Wwv-——-
.1»— JV-
Cn T c7, 7r- ole—If: Clef
F16: - ‘6
HH-H Rn R3 R7- 8|
———JWWL———[——’WWL— ————————— ~—IWVW——e——’Wlfiv——4}
.i_ ..L. __
CnT CaT’ CIT 6'
RC: - \7
M F? R s2. ' SI
8 ' W— __________ W IL 1L
1: —l[
J
Cn -; C3 F: G2 GI

 

 

 

FiG— 18

36

‘3
a

"ize tn

' .13 .
'_ LAV H

...

‘lit
'7 AA

"1

C3

x
‘—-L
'-’

No.1, '3‘

ycle,*.vnicn was u--‘

a
w

 

 

.l
.n .. .K e
,3 r u
3 d O l
. O I a...
. . u t V «a
C.” t 2v 8
(L. 3p n e +c
D 0 TM 14 H.
t 3 ......W ..L .
e .L ...“ C 3
Am w 3 t 3
_. r e i
o 3 .l .rJ h.“ .,
to” In Or. 1P.) t nv
my a. 3, \) DJ «J “S
n I. u 7), L e V ...o
«I 0 4L 1 IT. Z , -
C :1. 1 H“ 92.. u
. Tu ) VJ /.\ 1:1...“ .1; + u
«C r m .-.; n3 _ ...w . . .T..
. .2. l l _ I. e r .1
... mu 1 «Q m P A u .
I c 3. . [K e _ P. .
o .l . v. c . n 3 t c.
R n... It _ .1. . 17.9 .l . c .
.l . 1 . 3. r t
x l \. J . U. _ c... 2 e n
S _ 2.. F. . D. _ 3 3 o .
e ,..i D... .. - I\ . .l H ”1 .l n U LL
1 C 1 u .. _ R l: A .i. m . S
C 1— Q «L ..L S 4.. 9+ 0;. .Tu 1.. rd
/ o a t. O F a
3 i ,, If 1.. D. 1 f t \l C
a 1 n _ mu _ n 3 -....r
. v _ o AJ 1 3 A. Z 3 O 3 Vv 1 1
v c . _ . t P : mu. 3 .1 l . b .L n...
t r, P t n E .I._ u ...t «J t I . a
) J. z!— " it: :4 .1 ( n _ r
. ALP. 1.5 r. L .1. n .l ..T H d .3 e .
e . :wa w ..v ., 4 3; n n V _ e
T , If _ 1.. -To 1 e m... a. . .1 . V
i 1. D ..H . z: 3 . . .l .1. G ...; . .l
P. i 1.3 A . . . e l k . t
_ IT IL. . c, I? V. A u m... 3 . i
I” J i 1.. Vs n C A _. .1 _ c ,_
h r 1; 3. 3,. TT 3 1 a; m” _ D
.vuu re _ 1 fl“ 3.. I. v . ..A o +nu . i
P _ u 7 ,7) . P \l 3 _
mi a ...L Pr- w. h-“ a; _ e 1!. 54 C v P _ n-..
n l. c l e S 13 h l \l
f L t l_ o d ..T I T (x t e v P. 8
IT ...... n0 .Wn: Y t I... S _ P. \ .1
no lo 1... i D A a 3 e .3 . d.
.o i 1. 3 n1 . 4 .1. w. I. C ..n ..D 3.
.1 i 3 1 a = 3 G U. A
a L n J 9 r 3 q _ .c 3 9 u
.1 n _ ._ z r l ..n . _ .1. n .n V
i. e P. P. .1. f t t 2 r
r 2., 1 e I x .o 2 l e
3 8 m3. an . n S H O. n S y Y. mi
C w“ .1. I n.“ _. «a A a. V; i a .5 ....m.

37

t3 the original Gracess anfi tug rsmainuar is Egan exoa nded

3r (n41) tLges, expanding

1-“
:3
C
"S
D
U)
O
"b
1..
(‘1
(Q
9
C
r
C)
I
O
m
P‘)
C‘-

x
"O

or n cycles, the remilnd r will be a

'3
C'"
( L;
"S
is
U1
0
F I)
L:
m
P
p
D

y22 i;

(f
*4

H

,)

(.3

(D
.

constant WulCL guy as egresented 0y 3 ragls;'

nittance

f) 1
r‘S
Ho
<2
F J
L
L
’C‘
O
i.-
:3
cf
m

50, aft-:r (n41) tiges, tga

becones,

Dy f1 1r: C3A)- Che transfer Eépittfilce of this natwsrk
is
Y12 : AnP- ------- (115)

5 B)
how, if we connezt 11 Luege IaQCer skrgcuures in par:

1121, tge tr nefér algitt “9? W111 (Gt 2: t;e same as desired
but 13 we 'ugtiply all Z12 KA) by iiffer‘mt c:n€tints tne
dagire; tr ns;%r €5_itt:nce ;35 C3 fded. 00, L: we ault'gly
' " 3 ,l I
112 a :y 22’ A1 , ------ as. tLBA we hfive,

n a A

O;
F).
H)
"b
k'u
*5
U.)
:3
C"
O
U
.3
b)
(+
,d
73
r-f'
(J
u
{T
\J
{‘1
(D
(J)
H.
w
(b
k“!
F
[U
[0
3
c+
(1“
i
(D
m
p
C*
r-
I
(D
:<‘
s
L
...:
U)
U

58

 

.1 ————— + _- Y} --------- - n—-—-— I 1 1 A \
V: ‘10 ; -‘u ; —L-. \A La /
K3 [\1 LX411
‘ +_ r. ' . . .‘_ 2 7‘ \ ' ‘7‘ -_‘ , 0 .j ‘.- ‘ '. ~ ‘ n .‘ -. 1 v . r, r I; \ a
-.qu cl; 'VJ quid .L.) .va up JlI’.qu€, NB multigl'f t.it3 trump“
1‘» i: z 4- ‘ x '3. v -‘ 1 ‘. a, '7“ . .- . .-.
isr a . pt MTJ: :3 a, + A1 + ----—T An :tC- in, ;:r 1191
-lro | .h. . 1
15.1 -tllsl
. - . D L i - ,. .. i- - .. .. ' “ A . 4‘ - ’_ ~ ww , _ -
connection oi tnci: hcuno;iu will grogaue tbs 0;;e su\rt 'ir-

cult ri“

(L

-1 L

admittance will

.0 . .. . . L . ."
ill l-’:\'lI-1LJ :x...tl

'an. - -.. A "‘ -.. ‘ J- .., n
it“ C38 (tue 91V3n X43) tit tL? tr uiier
.:20ls [12
"1
V

 

".— .. ' A. 'r. 4. .,, - ' J .4 r“- .',_ ..._. I
b¢glsr A3.U,l Cu 1 * out Que leu ‘ Ci tue 0;.Jt at: A}
"'"w ‘r 1“ , '.' , ‘ a -- ",3 ' -‘ ~~ "fl ‘ -
”men the v l43 3i J lb l-iJV, n» Lecigcu var" l¢cor

ious to c lsulxte. ;at if P is siwll, -4 e li;r gathvd awy
be apslici to fini out tudpa v; ass. ”a :nom,
Y12 = AK?-L -------- -( i&)
.L.)
. tr , ,.
= 1k?» ----- —------—--—----- ilsi)
‘fi 1
D“~“4 D P $--4 Do
l '1
But man P-—)c -~- ------ «(122)
ng : AHPK ---------(122)
—L A —
_§5__
Now if V3 coriir figure (13), tiieu for P—‘p, the
imped nee of Lie ,AUJL Can filenJGJ 3n--~C4 are much larger
compared to the resistznues Rnl 1, El ----4 , -ng the current

357

S
H
LU
<
K
H
[H
k
H
k.
w
(J
L4
(D
F
D
(L
"S
(L)
*4
(
{u

through the FEJlSLLuCSQ
snunt electgnce in series. no tn: vgltwfie LL tie input
of the network is e:u:l t3 tnat just to tne left 3i

- g

elastance $2. Lherefcre, tne velue of A3 is only influenceu
J

by the network r14ht to tnis saint.

voltu es acro;s tie JOCiUC 33: 39 and 37, 2c C‘l e ) tilt

_ /
31 is the slt‘:e t3 tie lsit of 33 end inped nce lootini to
the right of 12 is very l r;e in cogaerison ultu g 00,

133+
.13— 2t.- : P . - ............ (107)
U? 3”b7 —/

J 3 /

 

---------(124)

to
So, the current tnrcugn the snort circuited output
terminal,

I2 = $9? ___ ............. -(l2‘)

0
"S
[3)
[U
ll
H

multiplying equ. (123) and (134), we nave,

5:—A X :39 = P P J
cu:— —:- -— . — - PL-
El E G S ng2 :‘d 7

a/

or E? X E”) - ‘P . P _. P2
W". —: - ... __ -
I‘d 7? 1 ‘fl‘ n A ’1 h r"
l a} 3303 agog G2J30393
OT 39 9 A
.;.= E: ------------- (1:7)
El 33335235

putting ne VilJe of eQu. {123) we iet:

'f‘q « ~ ____________ Q-
01", If) 3 P3
E1 §§33315353

Putting toe value of efiu. (122) in eiu. (133) we hgve

 

 

z
AXPJ = P3
—‘-- f1 \ A
Do U1*3~1D383 ----------- (131)
or A : DO
3 r. .i - ,9)
J1J2ul~£03 ------- (lg_
50, in a generalized for1 he may write tnis equa-
tion as,
A = D0 ------------- (133)
n r1 '1 VI“ ~_
(J1"’Un—1I\Dlod ----- on)

how, we will solve a nunerical exengle by tne above
syntneois orocedure:

Let tne transfer function be given Dy

----(134)

 

Y12 :A P : P‘l
§%§% (P +57 (P 45? T? I6)

 

= Pil A
P“ 12Pf¥44PI43 ------------ (133)

we choose,

40

41

B'cp) ; l <P43)(P45) (P47) -------- (136)
3
: 1 p3 4 522 4 23.6P 4 35 ----(137>
3
hence,

 

- a 23 + 7P2 + 20-3 4 13 ----- (1 a)
Z22 : l” = l
is? 3.?5 +592 4 23.7? 4 33 ------ (139)
zi
2 4 i
.443? 4 l
1-23 4 1
.4P 4 1
.31 4 i
33? 4 i

.338

The network is Snown in figure (21). fine transfer

admittance is cfllculatad.

_ ------- (141)
P9 4 5P3 4 23-7P4 35

\NIt—J

 

 

Consioerin Y22"5:in
Y22= 2

3P3+ 7P2+20.3P413 ------------- (142)

l

3P3%5P2423.7P435
let us turn it end for end.

Y>2 - 13 + 30-?P % 7P2 + 3 P3 -------- (14:)
i 3 __
35 l’23 7P 43P3 4 P9

42.

The network 1 given by figure (22) Y12 is calculated

(1)

C : i 4 1 = .667 ------------- (145)
' 2 23'
The * ittenoe levels of toe networks are multiplied
respectively by
2X.oo7 bx.bo7

The finel networg of the parallel ladder structures

terminates by a one ohm load is snown in figure (23)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

RVH-l Cn C5 Cl C"
1, _ _____.|L _ Jl . Jl
__M/\'—h I 7| N
a. a. a2 a.
F 4'6 —20
Rn-H Rn RI... $3 2,. I
- IL 1| ll
——’W———4/\/\r\-—— M 1F 1' TI
:E C n :J: CL, 63, Go. GI
FiG ,. I?
.008 x '6" "2g '5
——’M/\ We - W WW4 NVW—
AF 33 4 T: . (+1457
7i

Fig-2/

44-3

 

 

 

 

1,444

 

 

 

 

 

 

 

 

 

 

- 0042 -076
. .6/5 -3 Z
—4vw\M, W 44
. I ‘
== 63 :2 2.22 2.7
Pia-22
- .0”; /.2,/ l. 6 .66 7
w W 4w;
WP 243 :: .3 d: .33¢.
.00; we 45% /-37
4444—. i
“.252 Z: 5‘88 Z: ' 5;

 

 

 

 

FiG —23

1.5

Modified Hethod with ?ew3r Elements¥(32

 

Let us consider any of the soove lecder networxs with

transfer ed itts nce,

D
This ladder structure is driwn in figure (34) in such

(0

‘Fx ‘4 D . r~ 4' ' 3 V- ' " 3 H'.’ " ~ I' ‘r "- ("r->1 w
a 4~y that the lirst two elcnent rs shown sno the Ooharb are

U]

L

shown by a iox 5. cut since , the network nay also be

P It‘
n
[\FJL‘H

reorese nte j by figure (25)

Now El is the volt tags across the points A b (across
parallel combinations of CH andi-n4l)- So, if we interchange
the elements Cn and Y'n(l) as in figure (25), there is no
effect on the voltege E out the short circuit current I, is
011.11 is .

In figure (25) the current may be given oy

11 : E ---------- “(148)
Rn4l
But in figure (26) the current is m1sn.ed to

11' = ssnp ------------- (142)
11' = Rn41’3nr --------- (1)0)
11 = 13'

RailCnP --------- (131)

FL) t ‘9." 8 EC 1'1 0 ‘4": ,

’K( -
12 ' E;

22 -------------- (133)
1£§) = 11'

 

 

 

 

 

 

 

 

 

 

 

 

 

 

#6

 

 

 

 

 

 

 

 

 

 

 

 

Rh—H
1' “——-“WW" 2'
:_Ch 8
‘ 2.
Fig —zl4
' Rn+l
L, 9” MM “
TE), 8 EA! :ILC/n AI.
2. 1,— '
WG-25
ch
2, II
it
_ E2.) 8 EA Rh‘“ A I"
2.

 

 

 

96—26

447

1 1 ~ - T, 1
wnicn nas the sage forn as Y4§41) excepting tne constant

multiplier. From thi; result it is clear that a sinéle

ladder structure may yield a transfer adnittance to the

332‘ x (1 4 5:148: ) ---------------- (135)

may" (- 1,..‘r.-. - o -1 'ID . - -, .
1.11s i.13.u'_1:’. ..-..o t.;-.r.ns 0-1 :11).- (c4) 1:--1011 die-ans

that the ne work needs only numbers of purallel leader

n

2

structires instead of n numbers used in tne previous svn-
So, if we consider the nodified figure (27), we

have

it)
I"
U”
j
'11
P]
I
I
I
I
I
I
I
I
I
I
I
A
...:
K.)
O\
V

I":(

H
mg
:5
....

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H
a."
DJ
,4
$3
.1.
(.
:5
*U
I

I

I

I

I

I

I

I
A
H
.4
V

 

where a and b hesetne value between C to l cosoaring equ.

(157) witn (153) we get

 

 

or = A: X Rn‘icn --------------------- -(159)

 

O’ISD

.. .-. 0‘ , .- '4..-.. m . A. .~'
One more res1stence or CiQaCltAflJe may be elinlnated

OJ
<<
LD
(1
U)
s:
i
:5
J
p,
H
I"
H
H)
(J lw
H
O
"S
T
H
H
Ho
HJ
() IS»)
H

1.8

t

. _ . 0..
C .Tu C e .l
n a u h S .. o
n r t A e 3. D n
. c .1 t .l P e t e
3 .3. S n .C 1. 3 O ....... V
.1 r .1 .3 d _ .. t t k .1
no 2v ML Ge 8 ..Tu v1 ) A ...
3 .Tu C I C w . e 4v 7-
r d 3 ..a O n u b .- 2 1..
[Dru u. ,_ U ~fi w r 01 L (U / .\ r
O {M .... Q. \n/ l ....v ,J
H.“ To 1-. 3 . .1 l 1 r.“ o... a...
t . r S e .9 2 .i r
n E 3 r. _ H Y w E 4 f
y .l .H w. i . e. +c a i 3 ..u . o 0
1L ..L Y. 3 ...3 q ...... S to . ....
ml“ «a +U n . : .4; s U. e 01 B
O A; ”we ..v w; n. ..u ,3 fly .nL lad
.Q +L a L. . 1L . .1: .1...“ ,L “41.
Tu a: r S W; m. n. . .. . 1
Q 39 4U t . J 1r; ll\ .Tu ,5 1L
.-.. -A a. . 11 .. . . ... ,. 2|.
4r. I. . L (Q W. .4; m .. . 4L 4. A. .
4. 3 a. .l U. T 6 C .h \l \l w
e m T .C n 1 w _ r .. . n A _ l 2 \1
l r J E A. .l . C H n C 1. r? /J 7 , ... .
4.. ..IL _ .....w ....H 91 _ v L v ‘ u 1 l ffiv
C C r. l ..u w. J C .5 (\ z . 1 Q
q. . n.9,. H. «C 1 no. .1 u a... \i/ — — /|\ m.“
c 3 e .. . n_2 r 1 r .c t .... O . . . t
...... T .1 3 o ._ .. r0 . _ . 3
S 3 w. w; e a. win u a o , t U a; 1 . _ . ...U
1 ...... . . n a . u . . i . _ _
.1 n. to 9 .o w .e ... ..J. . . . _ J
t 3 e t .-.. .7. C ..c . u _ . . . ...
.l C ..l. m. . a.-. e . J 1. . . _ . t
.1 T n ... 3 I. ...... 8.. ... . n . . 1/ .
e V a. .1 .... t a L e 3 . 7. a . a... ..i
..I v l. v. 1.. 1. .o T _ n . 5.... a .c
Q P n e 3 3 .r. J c 3 _ a u v.” 2
. a C .1 L . 3 t .1 ....“ . . e
.1 1 “ .-. 1 C 3 .o 1. c _ 1L. Dy 1 D.
f 2,. . ...“ ... ...... . . r . u. 1 IT A-.. : a...
2 ...; .. _ C C 3 L .. . r a; 1 1 n C = H
9V «. . u .1” .11 .. n. . fi; 91.. Cu 0 ’4. ...v Q». ...,
h ... V. .1 e r .. . t ....u = ,o
t -_ t -L 1 W 3 m .. A 1.. h o i X K A . _ B 3
..4 e .1. ....” u. ... l 7. 1:1 . - . u. \/
fl 3 ... .2 T u . .2 5 .1. K 1 1 C t 3
O I u n F . t .. 3 C : , . n:
r «I... o 1AM“ _. . a x. , AI: W .— 3 WM 1‘ 15L -— - — — _ ., H O (
F , . n .2 c w , ., ... L I l o - o c o
c 3 t o u. .. c ... A .1 _b e
... e e .1. ..., V t 8 .n .
a. . . W n s. ...... «1 oi. .. w... I.
n T 3 3 .... 8 3 . J
«.4 srv ’ 1 1) 4 ..rL. 1L “A are in
e 3 1. 1 e .To 3 a. A c a 1
d .... r n e 1 e .. a
n Wu“ ”I.“ 44L ..J y DA 3“ «u. . U
a i t ... _ n_2 .. . x s t . ... ..-.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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0v
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8 EL 113%: ::Q—b)Cn be“
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Rh-H Ran-H
6* 7:3: 63>
Fié - 28
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Filtzr Ee;i;n:

 

In order to illuatrupe tht may be azaleved with

-‘

curves of 2 3ix pole polyncnial 3(P) are shown in the next

.—. .- ‘ ' .~~ - 1 fl ..., '- - .. .-A , , .. fl: 7 ‘n -_ .
>:.e WulCL VP: t¢&en fro; tAd g:sters tuSULS of A. zaur10£

 

 

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.... . . ..
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V W.p . .. o 91......

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..6'L.. .o 06 F666“... v6.u¢o....o..§ .6 v 6 . .. . .

.
.O-loiofar.ot6v.*-9nto .ovdol*cov.

.A6oo

 

 

 

 

60'06C.-- 066'.

.
.

 

 

...-6"}. 9.696 ..... do,YJ¢6.¢-¢.V-v'h

..c...166v.4 ¢?I.

T4<6+.. 6916419616
w-owEO. Oo-Vlfldl‘
vc¢4¢4f+u 6.4..469 Ti 0: O

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6964.9. .4?1o6.'.4 6 9
... 6067666.? 9*. ..9 '16. .
.1969666.+6<{.61*..W6.,.699. 9 6

.66.r6...?. .666 46:.
'19.. v . . . , 61.
. 6.9.6.69 +6.... W6 6T4 6.0
11-66 v-9.61_. . . . .... . . _ , .j‘f

146+

‘9- 66-9?
. .

144‘ 66".-
. _

198765 4 3. 2 I

53L

Develoement into a Lattice:

 

L
(‘f
('1'-
L:
b
w
0
k—b
W

Ehe :rivin; saint impedance or ad.

. - 'mm ’m “- A71- . . p. a. V“- .-_ ... . -.- . , .... ,
four ternine- “... netuerk “2e .angt an: 0: 23-,1ex frequency

Four terminal networks 32y be soecified conveniently

in terms Cf the Z or I co-efficients occuring in the two pair

of linear equations:

; I2212

P]
H
H
Hi
H
b3
,4
,4

----------------- (164)

[H
F;
H
P]
H
|<
a.
P]
[U
}<
H
R)

I2 : E1 Y12 6 E2 Y22

nd the

{D

If the netiork is passive then 212 3 291
driving point and transfer itnedence (and admittance) func-

tions may be exeaneed into pertiel fraction (43) and we he e,

Z11 = A1161» 6111(0) 6: all”)
P

53
54
[U

I

m
H-x
RX

a. '-

:1)

F40
m V
col-o
m
ea
R}!

b1
nu
n)

l

3

 

£53

l~<
H
H
u
0’
1—:
p4
l
V
to
.4.
U‘
H
H
C
V
.4.
"d
C7
’1

K:
H
N
H
U’
H
\f‘
"d
.4.-
C’
H
[U
C)
1..
H
r‘.)
H
[\J

l
[‘0
[\J

for all values of r

If sucn a four terrinol network is ;1"C:3 oetueen a
pair of resistances, one satiny as a generator und another

(L)
O
t— v;
(‘9-
L)
(D

as a loud. Then tne transfer impedinc complete cir-

 

 

 

 

HID]

The transfer iuoed;nce is a rational function of
complex frequency P Wit; real co—efficients (4). The zeros of
this function ere tne gene as/driang point function and
since the network contuins Only resiszences and cupacitances,
tnese zeros will be simole and occur at the real negative

axis of cunjlex frequency “lane. Ani, since the network

 

)K

 

 

 

 

 

 

x
K

-r

Fla —B‘

G 3 9
HQ —32.
II

N

 

 

 

 

 

 

FiG~33

."

contains PuSiSt3HC€S ml and R2 in series with two meshes, the

T"

U.

transfer honittance caniot have 1 zero &t P

dx.ninin3 tne equation of transfer function it can be

found that its poles are yroducec either byzeros 212 or the poles
of 211 or 2:2 which are not alreeéj containei b; the zeros of
‘ zeros of tansfer

212. So there are no restrictions upon the

function bee use there Inn no restriction uoon the poles of the

Q

J

tr nsfer impedance s72 excepting that they should occur in

l

conjugate wnen complex.
The typical position of poles and zeros are snown in

figures (31) and (32) resoectively.

*d
s:
d
d
H
:5
U 3
3
H
H
m
H
m
n
H
{<9
P]
{\J
H
O
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
H
_\'l
H

2? : El : g : l
I; I Y12 """""""""" (172)

This equation shows that only transfer inpednnce is
not sufficient to soeoifj the network because it contains only
one of the three functiois which soecify a four-terminal
network. The renuinln; functions Ill and Egg may be given
by any values, so long as the three functions obey the residue

theorem.

Let us consider here that the network is synnetrical

513

and tne elueli;; si'n of residue theoren holds here at each

pole.

simple and n; stive. Around this oole Zr'l may be ex;snded

into partial fractions of the fern of equ. (162) and the

(I
30
H

co-efficients no} he of any sign but r Now, if both 111
and Ygg are expanded sinilxrly into partial fraction, accord-

ing to our assunotions, we have,

that from the x; noion of qu : Z Ill and Y22 may be de-

rived only by taxinj 311 residue: :ositive. fhe aanittonce
matrix of this network is now conplstely :geoified end tnis
network is altiys realizable by n ans of a lattice structure.

The impedance matrix of this network may he found by

...- o

5j) and co;;nrinr it with equ. (154). From

_).
_‘.

solving equ. (l

equ. (163)

K
R.)
I u
H
I
[11
H
4-.
*<
N
ID
K:
H
tr;
to

F"

1 ' Yii Yes

$12 I2 : y122 E1

....
K
[0
{O
I-<
H
[0
[11
[U

oubtructing these equations, .e hove,

El (Y11Y22’Y122) : I>2I1 ' Y12 12

or, E1 : Yog w -Yl? I2
Yiiiss-1123 I * 411322-Y122

 

cogparfixrg'with ecnx. (164)

 

’ Y122 111122-1122 ---(177)

[\1
l0
R)

H

i5

H

The constant quantities are all , al2 (D) and

r -, .
a22 \53 of egu. (loo) may now be written as,

dim

P900 Z11 all (90) ------- (179)

dim
990° Z12

dim
B)ai 222

II
{o
§é

l

I

I

I

I

I

I
F3
(1)
O
v

II

n:
m
m

Now physical realizsoility Qeoon;s the; it should obey the

residue theorem

2
a(‘?{ a<w222 -a (fli2 >/- 0

be found re ;ily ;y the limits :gplied eoove t, the function

2 and 's reoleoed by Y's as given by equs. (175-173)

[‘3
DJ

22‘Z12

Hence,

CI 1, P.» w — ’— - {wA - , % “ ‘
“11‘ ) '22( ) ALL I - G*“_ 211 222-3122) ----(102)
P—yd7

= Lil Y1 Y92‘Y 22 ----(133>
Peav -l — 1‘

2

 

Tzllyaa‘Y332)

Lin 1 gggg ----(l$4)
P—M TYiiYes-Yi 2—" Y‘-

 

Ehis limit c;nnot be zero because then YllYQZ’Y1222D

Which is imocssible oeoeuse none of the X's heaoole at P23.

1

This oin be proved oy the act that Z : 0 when P c D and so

W

P = mend so blew” :0.

cf-

Y12 csn not have a 9018 a
“30' it “iv-S Specified that 811 (m) = 1322(3)

Q n ’_, A” o,
«9)- : 0. so, fron these we Chm concluce that

’fhe terns akfl) are the constant parts end by virtue

of the inequality, we can extrEct ‘roo each end of the network

a series resistance of magnitude

 

 

 

 

The network which is left is still physically

:57

re liceble because the only effect of extracting these

$0

(D

resistances will be to 3k the

I
,I

i U
r.
H
F“
(.
t“.
*3
I I
C I
h
I
(U
"S
F
I
It \
:3
c+
CT
(‘9

inefiu”lity sien hold for the terns a(D) in n function. This

\1

network, still synsetiricel, may now he realized

ohysic lly “s e lattice, the fires of which .3y be riven by,

N
I}.
II
N
a"
H
I—J
N
...;
O
I
I
I
I
I
I
+4
(D
O)
V

‘

This synthesis may he donellictle ouioker (43) o
rematoerin; that the resistances antto t; extr cted from
each end Encroxlnetely of the nsénituCe of the sssller series
resistsice of Z~1 I Z 2 end 211 - Z12, which are the era
impedenoes of tie lattice before the extraction and equal to
and (Yll-ylo) -' respectively.

To illustrate the stove procedure a numerical orob-

 

given
2: - ir‘i) (PI1) (PI4l - -------- (1)1)
<P-2) (9-31 (2-43
2P3it

 

263-24 ------- ( 93)

 

: -1 % DP3 4 3??
TEIJY (:¥3> (PIA) -------- (194)

 

Hi
"I” .
\N
*(I
4:

y = y; _ 1 I so? I 10? I 42F

11 ~2 1312? PT? W?

- 1] I .. '2‘: 4' 4'29 ) .-

L111. Y12 - 2 ( ‘T-P ‘ -
Prg 9T2;

 

 

 

{‘0

U)
I

 

II
H
+-

NH

144

N
314“
I

 

L‘ Q
U3 ’5“

 

 

 

'03 130

R¢ : 133 X 144 32: 71K71

b 71x3 13) i
R = l C; : 51

3 71 , 17:4

The network is shown in figure 35.

144

62.

 

 

 

 

 

 

 

 

 

 

 

 

 

FIG '35

£33

Develocseht into a stn'le lacoor strtctore:

 

We know that the transfer function of a network may

be given by

0

it is possible to find out short circuit drillinr point and

From the drivin; point function it is 31L¢ys possible
develoc a laiIer structure. jot in fjur terminal caseSthis

rninels

(u

should be developed in such a way that the gair of t
placed at the far end also realizes the transfer adsittance

Y12 (Or 212)

:3

he soles of both X32 e d $12 are conglex natursl

frequencies of the network and the lance” develobment auto-

natically gives a network in whicn the transfer function nes

procer poles.

The only thing left now s thatihow one can produce
the transfer admittance. In a la"der net-
work, it is possible to oroiuce zero of the tran'fer function
by making either the series branch ingedmnce infinity or shunt

' ~ - ‘, ' .4 ...-t .* .. ... ',—., r‘ --.~, . .\ ' r\ H ,'
or non llgcdzflce zero it the aggro riete 1P3,

condition 13 a n33€53ar¥ one (but not suff cient) to produce

‘- ~. , ' '1 'n- , .r & 2 it'- . , .: I .- . .. , -
T141 .3 ‘u'Ll L -‘J LAT ...; 91.x“: 1-; Ci :1"? L. a) 31.11 7'3 ('0 1‘. ’u t:.
1‘ A I. ' l‘ ’ ‘ " J. 1‘ v ‘ ‘.’ ’V ‘1 '- ’ " ‘ . l‘ .‘I
Ca'EZJir‘ed frUQU-BHSJ’ bLlr‘OJbL‘U‘Jt but} gay/114;-iCSlS Jr‘CDCBQSO

. 3 .x- - v. r". P .A 1 o 1‘- r‘ 1. I .~, ‘ --. ‘ r- v - -‘
a Hundrlcol exon,le will Snow tne whole trogedure

1‘ u— ‘,- 4- p - .v '1
Let L13 aogoUun’c} (41.1.3 iLlflCtlJ .3:

 

ix]
(D
' S
O
(D
O
*‘b
(“1“
1 ,
CD
0,
"5
H.
<:
H
:3
"0'
O
H.
h‘
*V
Cf
(
g
I i“
(‘7'
J
\ J
(t
h
o
I u}
(D
b-
H
}_J
( i
'-
’ l
" )
(U
"S
(D
c+
"5
SC)
H
5—)
(D
I

Let us calculate tne vslue of

If the sero freguonoj value of If; were is TEF than
that of the value uf Y2 at the point P : -2.5, we could shift

122 to the point P : 2.5 simnly bv subtracting a

V")

21 P00

N
(D

give real constant which is a conductance in tne actual

.JO
+

WOS

\

networ?. But tnis grocess isnot too? here because

Y22 (0) Y2: (fir-'3)

bo, we have to solve it in another way. fnis process

657

is to shift the zero at P : ~3 of er to P - ~2.5 and that is
through the subtrection of an oppropriete part of the pole at
P : '20

This pole 33v be given by the componert admittance.

............... ( 210)

3
a
re

ind out K1 by equating Egg (-2.5) with Y1
at the point P : —2.5, end by subtracting this value of Y1 from

I

$22, we can croduce a cero it P : 2.5

--------- (211)

ro

II

[D
Q

So the admittance to be subtracte from Y22 is

 

Thus the procedure shows that we have to build a
shunt bran h admittance of the value given by Y1, so that the
network at tne left of this point produce a zero at P = -2.3.

Ience the remaining admittance function is,

 

: ii343P2 + §.2p + 7)iiP % 2»il
(Pk) (PM) (5???

L-

C‘DV‘
LA

QVL

Now if we carefully ob ve t

"3T0

nittance, we find that tnis does not

the Y22' this branch

.3

F; L1; 8

V

n
.,1.

still contains

ero

snunt oranch ad-

CBC

in?
S;
(1)

l

(U

f:
w

0

*0

at P = -2 and tne sane is true for $12.se Actually in removing
the shunt briach Y1, we renoved only a part of the pole Ygg
at P : -2. But if the whole sole was reaoved, the branch would
have produced a zero at P = -2.

so, in the next steéc, we follow the same zero shift-
ing technique; but by removing a part of the adtittance func-
tion, we cannot shift tse poles of tie function. This nay be
reredied by the followin way. Through the application of the
shifting proCe‘Jre first to the ;iven function and the re-
sainder is then inverted and zero of the inverted function is
shifted, which is ectuslly the pole of the original function.
30, both the veros and poles of a function may be snifted to

any desire:

f.
A;

position rsgarole: t

originally were.
How, we tot lly renove the pol-

he inverted funct on because we have to
Y12 at this point. So we calculate the

this point.

 

7 -1
~22 — .
Yes
: P3 4 1-P2 { sop + 55
(.SifiPt J 5.25312.3S

L}

‘LI

critics

e at P :
or,duce

residue 0

1 p "\ ‘ Q
.L i Tog-JV

-203 fl”O'fl
a zero of
f 222' at

z A 3 (218)
.i \.
T,.

U)
II
H

01",

So, a series brgnch ihoedence of value

Ike P°I"Lfi er may we given oy,

In the above discussion 22 is a series branch, the

values of resistfince and cioacitance of which are given below

and this ‘rfinch :rodoces a 'ero at P : -:.3

The first cycle of toe network d-velopneit is
complete. Ye have jertially develOped th given E22 and also
we have been able to croouce a zero of 312. So, we follow
the 0’19 procedure to develoo the netw:rk conoletely and to

~J ’3'“. VI

produce all :38 zeros of Ylg

So, w- start the second cvcle by calculating the

, \ - . _ . t -
bince E22(bl (-2.3) is nevative, we cainot subtract

a positive constant (resJVIl cf conductance) from this value.
jut the zero shifting precedure may be continued

by partiel removtl of the sole at P : -2.8. This is done by

~

‘A l . -. ~ ~\ H 1. ‘ '- 1‘ 1 ‘ - I , ‘ -. . ‘ .__ 9' - ‘J ' I
suutractis, a ssunt orRJCL duuittance Y2. of tne $Lbu0d
/

Ziven eoove, the Vilue of snunt bramc; admittinse was found out.

§:§T3 ------------------ (323)

The velues of the elesents Lay ge given by,

 

22,( c) : IP46.3) (p42.sz --------- (224)
‘ .;-: (PIE-37 (945.4)

I

from wnicn a series brsnch impedance 24 may be sustracted

to prcCuce a zero of trenssission at P : -2.5 and Z4 may be cal-

I

N
.1:-
II
I33
I
I
I
I
I
p
K)
\J
\J

L.-

*I
I\.
U1

RB 031+ C =1.18

The sec>nd cycle is new congleted and tne remainder is,

(A
V
I
I
I
I
I
I
I
I
I
I
[0
I0
0\
V

222(d) 292(6)- Z4 : l_o(

Di-
4

\JIO\

"Ij v.)

r
o ’1'

LI
0
J“
\l
I
I
I
I
I
I
I
A
I X -’
m
.4
V

A
D,
V
I
A
*U
) +-

I
H
. _
m
*U Ul
.n'
C) i
‘\
V

0
VW
[A
("f
l
*‘5
K l
1,0
r-J
O
{.3
L I
3
C*'
A
"S
(T
L)
<3
_ I
'~
‘_
gu
"S
(D
U)
H
C)
(.1.
{‘3'
D
( 3
(D
V

51)
ct
by
U
N

R)

f0
t’)
I
...)
\i)

I

I

I

I

I

I

I

I

I
I ‘
ID
\J

 

---------------- (23:)
sinner of 292(d) :s
“3 : 1.? (P4 6.5) —.8 -----(:31)
P} 3.1?

 

r\
(I)

IO
[U

‘3 I}- n“ “\1.‘ a $' .‘ $— 4. ,- ., W .- .9, w .L- j-‘ —: .,. ,3}- , q.
(L— ht 014.2% V‘- L: bllkj' U U¢ad [.3 ‘)L U bL¢Lirl 'J v‘rd'alL/‘Aa. .LAI «‘IJ

------- (23:)

.. 43‘
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s, A.

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V
p.

1’3 7".
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LJ b \'
1

' ‘ ”In

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.. 1>J Uk‘s) I'Li
L ‘ + A -)
Q-¢~—~ \J U‘- J 78

(U

7/

3.. so;

 

 

 

$6

 

AN

 

Q,

 

fi

 

 

 

 

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swede.“

7|

is,

 

 

 

72,

 

.3
0-1 IU
AC

.5

”n:

r" ,
p..~ .1.

.3

‘ 4.‘ .‘
lg. L.--~r'v u

, V.-
v

_
k

u
I
l-

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-' ‘
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I

as

.../0

vs
Val
#04.

f".
lo

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IOIU
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~h0uld

”‘1
4
U

4- , .,
LJ.L’

eretor in

"‘ '7; I
'* V .1
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....u

o 41‘:

L

l“ ... ‘-1 7‘ '\_ A
‘1‘ ~.".‘ \J-d-l —~‘\.&

2 s

n, I.
qr“

n e t ‘I'I'

rj low

vs

.C‘

be 0

iis re-

-«-. .IIe ti

I.) 4'4
Irdrdh

“7
u

3 t-f3 re-

~1,~.

”I
‘1

”ll Ci:

:J L A—a‘

T)
\J

I
II
La,

t .

In

riction.

st

31

P

'-’\ .

.C

+ U
Av
...L

LL.
5

 

 

 

 

 

 

 

FIG -38
.‘L CI
W I
I K R.

 

 

 

 

§
§

Fig-3,3‘

E LI 1-7-—

71+

 

A . “ 6 ~ fi— ' 3 ~ 5 , \ r‘ -\
the Giulv Isms ?t ' l“? ‘ A1171 l. -io‘ - in o- t- 3 lust,”
(‘1 (3 c. ‘2?“ “‘3 ‘I' T ‘ ‘3 IX:\ ~23 "Ifl‘ .- fl _ ‘
4.1er _:;-DU«\A LJ U. .» «r \y, ‘4 “.3...UI.I.J La..-th ~X - ‘\l, V=U1 ’
1..
. A ’\
R2 = KR ?nd to : i
‘ r v“ A ’ ,, _ c] -_ -A --— o
[(0 IVW’ ii 5 asluctit-Ju? LJ’) .. lulu]- »~1.L« 1’31“th
tno anp eon ti,is, .3 .Cve
T - (IN - T- /‘.«."‘) 4 -:p (.- TT /""") a q ------ ( (‘7.r)
“1 I1 x Uli/ «IV I/ UK/ {IV «J h/v

1 ID (kR-J/HC)-I3 KR : o -------- -(:37)

U1 t1;
(*1 H
H
-I-
I
H
)4
\
u]
I
C-I
‘5
\
L J
V
I
H
..J
A
I
c,
h .
V‘V
\\
(’3
V
II
C)
I
I
I
I
I
I
A
[U
\N
U)
V

./

fl-I r I'— 0"

5L1 7 Io b3 - I7 ti; J no) _ O ----------- (2,3)
“- /

{KRj-(EBECR/N232) : (2K23/2‘I232)-(R/}f232) I J[MA-{303)-(ex-3R3/Js)
-(2KR//.Il:l) -(R3/II‘CU ------~(243)

43’5101’) it is 315.31" tllfit fit Li frequency

innere R - l B is real and use a manituie degendent upon

(wily K when I : 0.), thei 3 : O, wnicn is tne null point. gt
tnis: “oint, t“e network also ore uses an abrupt phase re-

verwsfil. I” K is le:s hen 3.5, tne°e is only a gertiel null

and network pnuse sniit attains a value of 1803 degrees as
is shown by tne real, negative value of 3. Phase reversal

occurs more grifiuelly es tne sloge of tne cuw'e cecreeses

’r ”I", “I-.- - , ; . .. I - - , ,_, ,‘ ,.. g
with n. lne onase Snift .nu JotBMtdstlun curVes are snown

in figures (4?) and (41) which are taken from tne paper of J.

'teined experimentally. The

(D
"5

2
' v

0
CI

Bower (6). lhe curves w

‘

res onse curve of suon netvork is snown in fig. (42)

An exssole is snomn "elom:

From the response curve of fig. (25) K : .25

OI‘,R=1
1

if we assume 0 = 0.1 G d then,

F
R ~ 1 : 1 5

...k *
-
—_

C)

= 3.2:»{104

7b

.30

.8.

so

 

I III JIVI

4.0.15. ...LIIIHI. _ .
f... LI. III “4 Th ¥o1c_L.. _fi...“.

.. m.“ hP+IT in
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. . . . . . . o k . . . u o . - r p o . . c . a ’ . . ‘ « . v . a . . o 4 . ¢ .+ . o u . . o . . * o o . . v o . 5 o o . . . o w . v ,0 p . . a . r o o 4 o ~ o 7 ~ g
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p, L >7 0 L 0.... r » LL F > FL F L Lr > L r P h
JJ 4 .t- 0 4 4 4 ‘4 4.1 H J 0 +444 ¢ fi4+ liifi 4 ¢ 4 . q -+ 1% 4+4 4
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o . . . . . . Q . . o . . . . o . . . h . . . . .c . v . + . . § . w . . . . . . . . “I . F . V o o o . o . . o w . . . o t . o . 9 o . o o o . o . + o . . .. c a L a W o_ q o o
.q . __ . . . . . y u -- . . . . . . + . . . o > . . . fi . a . . * . . . . . . . o 0 v . . a 4 o * o v . 4 o o . . + . o 4 . n . o 4 ,_ o o . u ‘ ,. . ._ v‘v a 0 A
v . . a . . . . . a . . . . w . ¢ o . A c . . . . . o o v . _. a N¢ I. a . u o o . y . u c + . c o 4 6 . . v 4 . o . v . . . w . ¢ . o ,9 ¢ o. . . . a . .‘A
~ o u . . 0 1 c 0 1‘ Av 0 A o . B o I. 9 0 4f 0 9 7|; :4 0| 0 -0’0 -LY :OI a v 0 ¢ x . 0 . o .A + .. a -r . 1. I . iv .071 -.0..,,0...l 10 . éfi 0. o! 0 -0 Jayl0; 0 I0! V .4Y lol-’ 9' cl ..1. 0'0: 0| 9 Arli‘l o- 0 0 1+ .14I 0 .1? TJY‘LI&‘I.OaI0I Iolxlvl. . . A
v ‘ . . ,. . . . . ,. . . . . . u . . . . . . . . . . ‘ . . . . . . . . . + . . u . . . . . . . . . 4 . . . g . . . v . . . . w . .o v o w a .y. o . v .
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m o . . . . . . . . .0 . 4 ... A . A. . o c u . . + . - ¢ . , . . . . . . V t 1 . o W . o - . ¢ . ,9 4 g t o v o A o o + . u u . * o v o O, L o p o 0 AV 0 o 4 o * A o ‘ r A
* . . . o . . . . . . . . . v . .H . . o . f . . : . * . . . . . . . . . . o . v . 0 o a * o c o v u o o 5 ,9. . . . a! v . o . . . u . 0 ¢ a v o v... ‘ o A
' l Elli». ‘IIIIII? [[1 r r I t. 6111.0» Fri. ..txsé {L - :17 P . r _ u. ¥ . r F , F

 

 

 

 

Band

77

Eliminaticn Filter:

 

very recently.

{24- D.

network shcaid be sy;netrical with respect

R1 = Q : K ------ (
'27:" ‘31
A:

From Stant3u s (61)

a=R
1:;

ulRl

jQ
II
if?
'50
H

(X
C2 " %{<R+R\) ::

where/a : 2777’; RC

R; bfiqg (3"- RI) ::

was su;3e; ed by X. ocno(47)
(3’)) weerefi?0 and
characteristic of the

to tne resonant

attenuation) frequency and

P = J g
fo
b - :r
.' 1 (245)

 

RX/AC’(|+K) (246)
c ONO/ALK (247)

(243)

f0 : Resonant frequency

The transfer function of tne netwark is

r13
'11
I O
4..
c.)
“d
4-
$11

calculated as

where F

G

= (may (MK) K?
:1. (I+A")(HK) (JR 4

e-(r+igi)

+fifi

iL

{L

is.

wow) K—
:(1+K) + (1+1;
$2 1%)

(252)

ametricel with rescect to

 

Since the re ponse is sy.
resonant frequency, we have,
3? {= E) }
3* P _: i P
“1 ,3" /
“1
80, F : H ------- 2 )A)
0P» (J+x15) f/ __
KR.: 'Ti
Let
'fi — TrfL 4- R.= 9(‘7
/&”=nKx-|
our transfer Iunctio ceccmes
E’; 2 J. PL+I - ..L '
E. M 5159+. _ M H-Af...
P P"+\
unere, M : |+n + (HR) hx
A. = J, <|+K) (“Wt‘x‘”) “x
tfl/k

nov: E}:
El 1 9:9 : 59 i : : _ 1
H1 Paw "L
Inicn i re resent: the leis at extreu:
0nd f2 ()f1) 5? the frequencies far wnich
32
3T” 1 ------ (:62)
3M

it can be snswn

(235)
(2.55)

A
[U
U)
K]
V

0‘ A.“ . H
- feq 12110.13 3 o

O\
k»!
v

C);

which is the bend wi tn.

(D
L
o
A
I .,
\2
k t
E;
L
(I)
d
U
(J

The roots of tne denozin tor of
real and negmive and so A) 2..

So, if M and A; are prescribeo the network elements
can be found outl But if it is desiraole to find mininun A

for ‘rescribfiifix cm'vice Verdi tnen fron equ. (253)

K : ®—\'Y\—h7L)/nx (254)

()3
I3
Q;
h
0x
Ui

\/

Atz-hX'L—r (M~I-—Y\)7(-’I (

or Azm 0-“)X+M+\;j_)_ ,, (- )
-nx +(M )— —n‘)>:'\
It can be shown that (3 he a positive minimum value

{\J
O\
O\

 

and.Xk2 is ”ositive for some positive value of X only if,

:1) (use)? ... (257)

The value of x for wnich IN is minimum is found,

X = 1&2 ~2--.1n¥ (l-nLC-i
A (1In) -(1-n)<

 

and A _ NW 0+r0 H. ..-(269)
fM‘1—(\+n)M-r t“)

minimum value of A is found

s- "J” _\
3'1: I+fl+2J—n A --' "- (:_7C./

.[A'bu
She relations (26?) and (2 m) are snown in fi 13. (43).
the v lue o of X, K and/q are Get-e gained as, xz—g-,K=g(q1<= “K1"
a ’nL :— nA:+:+Q/'A +2.
M+flnM+Q ) "f;
‘L
iV1(l+n)—(l’n) : 0-” + 2m ('+N)rA—%—Lfi

t ‘L
1%[M15-ZM +Q—n)] : 2 %;:—3-+ 9&fi91—Ln

where 0Q

‘\.Fb
ll

low:

('1)

Q . . - . V
An exanole is SLan :

Letf:33 A=3

From tne

SDlr—J
I

N. -4\TE> 52
H II II II

P4
u

(H;
*3
(3.
k} .
D’
5?
(D
(D
G
U)
(D

.3: 4.4

M (1+n) -(l-n) - 3 2(1.435)-

1 ,

fi' m--91 4 (l-n)€] : 2.2 3-5.4 % c,§] =
6* 7.9

—’ = -———: . 8

,3 #J* I 6

'f 84; - L/é

i=7?“

Ex
(I
3
'X
X
.r‘
{I
Z)
l
(I
-Q
0
\J

3 — .Lifi , a(
K- : .0086/(1/4 C2,” C/‘(tK -—- 04flf .
- K

.§
’ (Wm-2 M” cam.

.. f '
:LZMXlOg 1L: fl‘f‘:z.fi}‘\° M

L A
1 _ _

yea...
9.1141”
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c
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.

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7.. O r\.n:.

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1.1 .

j‘i,’9.1. I’ll ..010 fig; .Q‘zltlocodh‘i i016! ‘41.:(0‘1

1 . _ .
Q...O-"Y.a'. 1. 5+d..1¢ilo-o1.0 ... ,ol.o...~v4n'b on

m). ch.noo ooV*u 9L.» >Avhov10.

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1

11 .
its

.177cb-c. ottoaqcttt .1

0t .05. O. 'o‘CoO.1cOooo so+001.v.-QP'000
.

Jithfl’. .‘o . '1“

13v10¥~11 O.tn~.)¢v1¢l-<,
— A . v .
onioOTLOJ'F“. vfiv‘ob'u A..- Xovfiuv. {6.00.0141-
1 _
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Tw.o¢v0. ...-xtjv1‘.29. .-
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1
O.nivv4,ofio.b- O o‘IO
.

1
Yotn,t1! .YoIQ.-|. o

rthtft.r1.lw<wlo. . . .uwfol
.MxopWOTOV .10... .vatco.
cvoJ9§nvoto1n 1
c¢04u9v7§41

r .

 

86‘

.\.

V

. syn-
.J.M'.;..J.

1,,
111 ~\. ‘13

~

1’. 1‘; L1. 1

1. 11"

I

3

fi-

1

r .1 1 I1 1‘

‘ '1
J

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LiLfl .

1.. D S trt:

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VAL;

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‘-

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0
e .1.
oz. _ S o S 1..
O S .‘L _ «U ...H“ t urn 1.1-1 H
1v ..b n ..-“ _ -TV r .11.. 1L
e .l 3. vJ W e .l C 3 ,1 .1
S n p r w w K x p
u 3 1 .11 f . u n t r 1-“ -.
I 0 w.\ T u . 3 C ..1 1d,. 1 .1 1.
t +u «1 O on. VJ S .71 n .1.“ 11. .l
J C ‘3 .1 Wu 0 r wt. ml 0 .TU my 0...
J 1.: .1 .. .7. n... 8 S e .... U. 7
“.1 l t a, n 1.1 t u. 1,... 1.1 n r
t L ..1 e n 3 u l t n f 1..
i t ...». .l t .l a. r ... 1 n 1 1.
v... H 3 1 1b a... n .-1 1b 1.11 3 -
. .1. AT. 3 C +11 ..- 1
5 fl ....1 3 3 11. .1 3 T . C C H
.1 1Q 3 .l d. .1. V .111 L :1. 1 [a ..-.
~D .111. ...“ .1. 4 hi 41¢ 1 t o m”. ._ 1. .1
«.3 «J A v ...g v.1 ml. .....n 1.. ml. 1 1 AC 11V
M. /O 8 f .) 1L 1 d .l . C 1-1 .C F
TV 1 D 0 fit, Q .Tu 11.. C 11.. vi .1. u. 3 a
n 3 1 ,1. b .1. 3 a.-. C
...J .. T S T O .1 11. 3 T e . C
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