CHAPTER
3
OA-C FILTERS ; DIRECT FORM SYNTHESIS^
3 .0
In trod u ction
This ch apter i s concerned w ith the r e a liz a t io n o f
secon d - and h ig h er-ord er OA-C f i l t e r s , using the d ir e c t form
syn th esis approach.
This syn th esis technique should be such
th a t i t r e a liz e s the fin a l network in the form s u ita b le f o r
the m icrom in iaturization in the MOS technology^® ’ ^ .
This
im p lies that the c i r c u i t s should con ta in only the a ctiv e
d e v ic e , v i z . , op era tion a l a m p lifie r s , and ca p a cito rs in t h e ir
70-7^
r e a liz a t io n s
. Moreover, a l l the im portant parameters o f
th e network should be a fu n ction o f ra tio e d -C s.
Three techniques are proposed here f o r such a purpose
and are c r i t i c a l l y stu d ied .
These are ;
i ) L-Based Approach to OA-C sy n th esis ;
i i ) FDNR-3ased Approach to OA-C s)Tithesis ; and
iii)
FDNCAP-Based Approach to OA-C sy n th esis.
In any d ir e c t form sy n th esis technique, th e s ta r tin g p oin t is
the p assive RLC-prototype network, whose elements are to be
r e a liz e d in a manner so that the fin a l r e a liz a t io n contains
only those elem ents, which are s u ita b le f o r convenient m icro­
m in ia tu riza tion .
In the presen t ca s e , only the OAs and Cs
t The m aterial presented in th is ch apter has led to the
p u b lica tio n s o f the au th or’ s paper no. 3, and k l i s t e d on
page v ii.
8 8
89
are perm itted in the f in a l r e a liz a t io n s .
The r e s t r i c t i o n
on the use o f only OAs and c a p a c ito r s requ ires the sim ulation
o f two d if fe r e n t types o f c i r c u i t elements in each o f the
th ree proposed techniques.
The a p p lic a tio n o f a p a r tic u la r
technique w i l l la r g e ly depend upon the c o n fig u ra tio n o f the
p ro to ty p e .
Each technique w i l l be d iscu ssed in d e t a ils in
the subsequent s e c tio n .
Here we only presen t t h e ir o u tlin e s .
0 In the L-based Approach to Oa -C S y n th e sis, the R and L
elements are sim ulated and d ir e c t ly su b stitu te d in the p ro to ­
type to giv e an equivalent r e a liz a t io n in the OA-C form.
0 In the FDNR-based Approach t o OA-C S y n th esis, the admittance
o f the p rototyp e network (N) i s f i r s t sca led by the complexfrequency v a r ia b le (s ) to g iv e the transform ed network (N ^ .
The transform ation re s u lts in con v ertin g an RLC-network in to
a corresponding CRD-network^^.
The f i n a l r e a liz a t io n i s
then obtained by sim ulating the R and D elements by the OA-C
sim u lators.
I t may be noted th at the tran sform ation does not
a f f e c t the tr a n s fe r f m c t i o n o f the networic.
0 In the Fl^CAP-based Approach t o OA-C S y n th esis, the admitt­
ance o f the p rototyp e RLC-network (N) are sca le d by the func­
t io n , s^, to give a corresponding CDF-network (N^^), without
a ffe c t in g the tra n s fe r fu n ctio n .
The fin a l r e a liz a t io n b o ils
down to the sim ulation o f D and F elements in the OA-C form.
For a l l the th ree tech n iqu es, the OA-C sim ulators,
90
already studied in Chapter 2, are used.
As the sim ulation
o f two d if fe r e n t type o f elements g e n e ra lly requ ires a
la r g e count o f a c tiv e and p a ssive components, methods are
a ls o suggested f o r the op tim ization o f component count.
The three approaches are c r i t i c a l l y stu died f o r the
r e a liz a t io n o f secon d -ord er and h ig h e r-o rd e r f i l t e r s , res­
p e c t iv e ly in S ection s 3 .1 , 3.2 and 3 .3 .
Suggestions f o r 't h e
op tim iza tion o f component count and s u i t a b i l i t y o f stru ctu res
f o r a p a r tic u la r technique are c l e a r ly examined.
In S ection
3 .^ , experimental r e s u lts on some c i r c u i t s are inclu ded.
The
con clu d in g remarks are given in S e ctio n 3 .5 .
3.1
FDNR-Based Approach t o OA-C F i l t e r R e a liza tio n s
In the Frequency Dependent N egative R esistan ce (FDNR)-
based approach t o OA-C s y n th e sis , the p a ssive RLC prototype
c i r c u i t d escrib in g the tr a n s fe r fu n ction i s f i r s t transformed
in t o an equ ivalent network by s ca lin g each admittance o f the
o r ig in a l network (N) by the c o m p le x -v a r ia b le (s ), as shown in
Fig. 3 .1 .
The new network (N^) i s obtained by con vertin g a
r e s is t o r to a c a p a c it o r , an in d u ctor t o a r e s is t o r and a
c a p a c it o r to a frequency dependent n ega tive re s is ta n ce element^^.
The FES^R has an impedance o f the form , 1 /s D, where D is para­
meter o f the element w ith the u n its o f square farad-ohm
(f , ) .
Thus, the RLC-network (N) i s con verted to a CRD-
netwoiic (N^ ) without a ff e c t in g i t s tr a n s fe r fu n ction and the
91
NETWORK (N)
Vr
o--- V nAA— o ^
N E T WO R K (N )
5C‘
II
o;
VsL
o----- -o
sC
o-------- II------ o
Fi'g.3.1
c=Vr
Vr'
■AAt^----- o i R = L
Transform a­
tion
C,RD e q u i v a l e n t s of R L ^
s^D
elements
92
sy n th esis problem b a s ic a lly becomes that o f the r e a liz a tio n
o f r e s is t o r s and FDNRs in the OA-C form.
I t may be noted that in t h is tech n iqu e, the fin a l n et­
work r e a liz a t io n requ ires the sim ulation o f two d iffe r e n t
types o f c i r c u i t element, v i z . , FDNRs and Rs.
This can be
achieved by e ith e r rep la cin g each component by i t s id e a l Oa-C
immittance sim ulator or by re p la cin g a s e t o f components by
the equ ivalen t n on -id ea l OA-C component sim u lators.
Generally
the c le v e r use o f the second approach proves out to be more
a t t r a c t iv e from the co n sid e ra tio n s o f minimizing a c tiv e and
p a ssiv e componenxs and in the r e a liz a t io n s having a tt r a c tiv e
s e n s i t i v i t y p ro p e rtie s .
In th is s e c tio n , the FDNR- based technique i s used f o r
th e r e a liz a t io n o f ( i ) standard secon d -ord er OA-C f i l t e r s and
(ii)
h ig h er-ord er OA-C f i l t e r s .
In S e ctio n
a standard
secon d -ord er OA-C band-pass f i l t e r i s r e a liz e d by employing
id e a l FDNR sim ulator.
This i s follow ed by the r e a liz a tio n s
o f standard secon d -ord er OA-C band-elem ination and a ll-p a s s
f i l t e r s using the n on -id ea l FEWR-based component sim ulators.
The advantages a sso cia te d w ith the use o f n on -id ea l sim ulators
are c le a r ly demonstrated.
The technique i s a ls o employed in
S ection 3 .1 .3 in the r e a liz a t io n o f h ig h er-ord er f i l t e r .
93
3 .1 .1
R e a liza tio n o f a secon d -ord er s e c tio n s w ith id e a l
FDNR-simulator
The importance o f standard secon d -ord er f i l t e r s e ctio n s
in a c tiv e syn th esis hardly needs any e la b o ra tio n .
In Section s
3 .1 .1 and 3 .1 .2 the r e a liz a t io n o f secon d -ord er f i l t e r s , using
th e FDNR-based approach, i s demonstrated.
These employ the
FDNR-based sim u lator o f Table 2 .2 .
The r e a liz a t io n o f a secon d -ord er band-pass f i l t e r i s
f i r s t demonstrated by employing an id e a l FDNR sim ulator.
C onsider the p a ssiv e p rototyp e secon d -ord er band-pass (BP)
f i l t e r , shown in F ig. 3 .2 ( a ) .
I t s s c a lin g by the complex
frequency ' s ’ leads to the FDNR-based v e r s io n , shovm in Fig.
3.2 ( b ) .
For the fin a l r e a liz a t io n o f the BP f i l t e r in the
OA-C form, the elements in the shunx arm are to be replaced
by the appropriate component sim u lators.
In t h is ca se ,th e
id e a l FDNR i s replaced by the id e a l OA-C FDNR sim ulator o f
Fig. 2 .7 and R;C shunt-br^nch i s rep la ced by the immittance
sim ulator o f Fig. 2 .1 8 .
in Fig. 3 . 2 ( c ) t .
The com plete r e a liz a t io n i s shown
D ire ct a n a ly sis o f the c i r c u i t , y ie ld s the
tr a n s fe r fu n ction as :
^
'’ P
w
^
°
)S
s ^ + (^ )s t w 2
w
H, „ ( t ~ ) s
= _&E_9-----
(3.1)
D(s)
As an a lte r n a tiv e , the r e s is ta n ce o f the shunt branch may be
replaced by id e a l R -sim ulator o f F ig. 2 .1 6 . This however
req u ires matching c o n s tr a in t and la r g e component count.
9^
-TO BE SIMULATED
+ o-
t
\
_ ii
1
''i r, C - R ^ | | D ±
- o
L , ‘__ I
l__
-o -
(b)
(a)
REQUIRED O A - C SIMULATORS
(C )
-o +
OUT
---- o
IN
o-
IN
o—
OUT
-o
_o -
(d )
Fig. 3-2 (a)
P a s s i v e R L C prototype b a n d - p a s s filter ( B P F ) .
(b) F D N R - v e r s i o n of p r o t ot y p e BPF.
(c) O A — C v er s i on of F i g . 3 - 2 ( b ) .
(d) C - M O S inverter.
95
where,
and Q are, r e s p e c t iv e ly , the mid-band gain,
the p ole-freq u en cy and the p ole-Q o f the BP f i l t e r , having
the follow in g r e la tio n s h ip s :
C-.C,
1 /2
]
^
ol
°
° ^
and
C,
B„
Q =
^
C^ . Cj
^ ol
where,
C ,.
^k*
1 /2
j
(3 .2 )
output has not been taken at
the output term inal o f an OA, the response w i l l be load
s e n s it iv e .
In order t o ob v ia te th is problem , i t i s recommen­
ded to use a b u ffe r stage co n s itu te d by C-MOS in v e rte r
shown in Fig. 3 .2 (d ) .
An im portant p o in t t o be noted is that
the derived c i r c u i t may p ro v id e , in a d d itio n ,
standard
responses
the r e a liz a tio n .
at
the
other
outputs o f the OAs employed in
For example, the BPF o f Fig. 3 .2 ( c ) r e a liz e s
standard secon d -ord er low pass response a t the output o f OA^,
w ith the tra n s fe r fu n ctio n given by
T, ( s ) = ^
V.
"°D(s)
(3 .5 )
where, the low pass gain
=
(5 .4 )
From (5 .2 ) to (3 .^ ) i t is seen that a l l the f i l t e r para­
meters are in terms o f c a p a c ito r s r a t i o s , which is a
highly
96
d e s ir a b le fea tu re fo r the implem entation o f the c i r c u i t
in MOS-technology^^"^^.
The c i r c u i t p ossesses independent
tuning o f the p o le -fre q u e n cy
by b ia s -v o lta g e c o n tr o l.
The p ole-Q i s independently tunable by c a p a c ito r C^, without
d istu rb in g the p o le -fre q u e n cy .
The s e n s it iv it y study o f the f i l t e r was a ls o ca rrie d
out.
I t y ie ld e d the s e n s i t i v i t y fig u r e s o f f i l t e r parameters
o f in t e r e s t , as given in Table 3 .1 .
The ejqiressions c le a r ly
demonstrate the s e n s it iv it y fig u r e s to be a t t r a c t iv e , being
no more than unity in magnitude.
However, i t may be r e c a lle d
th a t the r e a liz a tio n o f id e a l FDNR sim ulator i t s e l f requ ires
c r i t i c a l component matching in Bs, i . e . ,
= B^.
This i n -
turn makes the o v e r a ll r e a liz a t io n f a i r l y s e n s itiv e to B2 and
B^, which i s not evident from Table 3 .1 .
The study o f the s e c t io n demonstrates how a second-order
OA-C f i l t e r may be r e a liz e d using id e a l FDNR sim ulator.
The
problem a ssocia ted w ith the use o f id e a l FDNR are ; ( i ) the
c i r c u i t s normally tend to re q u ire a h igher component count.
The present c i r c u i t uses fo u r OAs, f i v e ca p a cito rs and a b u ffe r
and ( i i ) the id e a l sim u lator req u ires matching o f components
f o r i t s r e a liz a t io n , making the o v e r a ll c i r c u i t s e n s itiv e to
such parameters.
97
98
3 .1 .2
R ea liza tion o f secon d -ord er s e c tio n s w ith n on -ideal
FDNR sim ulators
The s e ctio n demonstrates the a t t r a c t iv e featu res o f
f i l t e r r e a liz a tio n using the n on -id ea l FDNR sim ulators. The
r e a liz a t io n o f two standard secon d -ord er f i l t e r s are con si­
dered as examples, v i z . , band-elem ination (BE), and a ll-p a s s
( aP) f i l t e r s .
The d e t a ils o f the r e a liz a t io n and study of
these f i l t e r s are d iscu ssed as fo llo w s ;
0 Band elem ination f i l t e r - The p rototyp e secon d -ord er bandelem ination f i l t e r is shown in Fig. 3 .3 (a ) and i t s transformed
FDNR v ersion i s given in Fig. 3 .3 ( b ) .
To r e a liz e i t in the
OA-C form, the shunt brancn i s to be sim ulated.
The n on -ideal
FDNR sim iilator o f Fig. 2 .1 0 has been used in the d ir e c t simu­
la t io n o f the e n tire shunt arm (shown w ith in the dotted lin e s )
o f th e transformed network.
The f i n a l r e a liz a t io n o f the BE
f i l t e r in the OA-C form i s shown in Fig. 3 .3 ( c ) .
The analysis
o f the c i r c u i t y ie ld s the tr a n s fe r fu n ction as :
It, ( s ) =
^
3 =
°
( 3 .5 )
where
C2
1/2
^be ^ ^o^^ol * '^o ^ '^n ^ (% B 2.
and
Cn
Q = (1 + ^
)■( - ^ . - ^ )
1
\
^23
1/2
(3 .6 )
99
Ro
-0
+
R'
ci
Vf
Vb
(a)
r
-N O N -ID EA L FDNR
0 _______ It _
/
/
/
R
7
/
/
± c'
D^
T_
/
Vb,
/
z____________
(b)
r - O A - C SIMULATOR OF
N O N -ID E A L FDNR
o-
-t-
r*
cv
C2
V lp
Vbe
"TT
(c)
F i g . 3.3 (a)
Prototype-RLC band-€lemination
filter(BEF)
(b) S e a Led v e r s i o n of B E F
(c) O A - C
v e r s i o n of
B E F of Fig 3.3 (b)
100
N ote, once again the f i l t e r parameters o f in t e r e s t are having
the a t t r a c t iv e featu re o f being in the form o f c a p a c it o r r a t io s .
The c i r c u i t en joys the m u ltifu n ctio n a l c a p a b ilit y
and p rovid es two a d d ition a l standard responses o f band-pass
(b p ) and low -pass (LP) a t the outputs o f OA^ and OA2 , respec­
tiv e ly .
The tr a n s fe r fu n ctio n o f the BP and LP f i l t e r s are :
D(s)
and
T
(s ) = ^
= J^_2_
D (s)
(3 .3 )
where, the re s p e ctiv e f i l t e r s gain are given by
Hbp =
, and
I^.p
(3 .9 )
^ol
These are a ls o in the form o f C -r a tio s .
O bviously, the e^qare-
ssions f o r the w and Q remain lanaltered f o r the a d d ition a l
o
respon ses.
The c i r c u i t p ossesses independent b ia s -v o lta g e tuning
o f p ole-freq u en cy w^ (= w^^).
A lso, independent p assive tuning
o f p ole-Q i s p o s s ib le w ith
a n d /or C^.
The s e n s it iv it y
fig u r e s o f the c i r c u i t were d erived out and the f in a l r e s u lts
are inclu ded in Table 3 .2 .
I t i s seen that a c tiv e and p assive
s e n s i t i v i t i e s are, once again, w ith in unity in magnitude. Note,
101
■I
fc
t
. J
102
high-valued r e s is t o r s are requ ired a cross c a p a c ito r s ,
and C^ , to p rovide dc b ia s in g to the in v ertin g -term in a ls
o f the OAs.
0 A ll-p a s s f i l t e r - The secon d -ord er p rototyp e o f an a ll-p a s s
( aP) f i l t e r i s shown in Fig. 3 -M a )
sion i s given in F ig. 3 .M h ).
and i t s transformed v er­
For the OA-C r e a liz a t io n , the
com plete shunt arm, shown w ith in the d otted l i n e s , may be
sim ulated by d ir e c t ly using the n o n -id e a l FDNR-simulator o f
F ig. 2 . 3 .
Fig. 3 . 4 ( c ) .
The fin a l OA-C v e rs io n o f the f i l t e r i s given in
The use o f n on -id ea l (FDNR) re s u lts in an appre­
c ia b le op tim ization o f a c tiv e and p a ssiv e components.
A nalysis
o f th e f i l t e r y ie ld s the tr a n s fe r fu n ctio n :
where.
^o
a p p ' o
'"ol
o
S
1 2 p
^23
^4
p
^45
and th e AP-response i s r e a liz e d by s a t is fy in g the component
c o n d itio n ;
C
C
1
C
a-, = b. = - ^ , when -2 -= - ( - 1 - 1 ) .
r*
p
r
^23
^1
^2
(3 .1 1 )
This c o n d itio n can con v en ien tly be s a t i s f i e d , p a r tic u la r ly i f
the c ir c x iit i s implemented in m on olith ic form in the MOS
tech n olog y .
As in e a r l ie r ca s e , the c i r c u i t has m u ltifu n ction al
103
-JVy/v-
-o
Ro
Vi
+
1
Vo
Ri
-R
C
-o
(a)
— N O N - I D E A L FDNR
1
Co
- S I M U L A T E D N O N - I D E A L FDNR
------------------------------------- 0 +
-\h -
C3
'+
T
Vip
-»l-----C4
C5
Vbp
(C)
F i g . 3-4 (a)
Prot ot ype
all-pass
f i lt e r ( A P F )
(b) T r a n s f o r m e d F D N R - v e r s i o n of A P F
(c) O A - C f o r m of A P F
Vap
104
c a p a b ilit y and p rovid es a d d itio n a l standard secon d-order BP
and LP responses simuiltaneously at the outputs o f OA^ and
0A2 » r e s p e c tiv e ly .
The v a riou s p e rtin e n t parameters o f LP
and BP responses are given by ;
Hip = f^ol - V
“ - r^1
C.
= (B,B^. ^
■
C,
)
1 /2
.
(3 .1 2 )
and
U = 2( 1 +
Co
Cl
B?
~
B^
C,
C,
C25
)
1 /2
.
( 5 . 12 )
Once again, the f i l t e r parameters have the a tt r a c tiv e fea tu re
o f being in terms o f C -r a tio s .
v a riou s
filte r
The s e n s i t i v i t y fig u re s f o r
parameters w ith resp ect to the
a c tiv e
and
p a ssiv e eleraenxs are analysed and the f i n a l r e s u lts included
in Table 5 .3 .
The ta b le shows th a t the c i r c u i t enjoys a ttr a c ­
t iv e s e n s it iv it y perform ance, being l e s s than or equal to
unity in magnitude.
The c i r c u i t a ls o enjoys independent
e le c tr o n ic tuning o f p o le -fre q u e n cy .
This f i l t e r can r e a liz e
high-Q values due to the a d d itio n a l m u ltip lic a tio n factor^
(1+ C ^ /C i), in v olv ed in the exp ression o f (A.
To p rovide the
dc bias at the in v e rtin g -in p u ts o f both the OAs, high-valued
r e s is t o r s are required a cross the ca p a cito rs
and C^’.
I t i s important to n ote that both the c i r c u i t s , re a liz e d
w ith the n on -id ea l
based sim u la tors, have very a ttr a c tiv e
105
106
s e n s i t i v i t y p ro p e rtie s .
In ad dition ^ th ey norm ally do not
req u ire component matching in t h e ir rea liza tion s'*’ , and
t h e ir t o t a l a c tiv e and p a ssiv e components are con sid erably
low er.
This i ll u s t r a t e s th a t, whenever p o s s ib le , c le v e r
use o f n on -id ea l sim ulators should be made in the r e a liz a ­
tio n o f networks based on the FDNR approach.
3 . 1 .3
R ea liza tion o f h ig h e r-o rd e r ladd ers
H igher-order f i l t e r s are freq u en tly r e a liz e d in the
form o f lad d er stru ctu res^ ^ ’ ^ ^ I n
t h is s e c t io n , the FDNR
based approach i s applied to the a ctiv e -C r e a liz a t io n o f an
n -th order a l l - p o l e la d d er.
The v o lta g e tr a n s fe r fu n ction of
the LC la d d er, shown in Fig. 3.5 ( a ) , i s given by
T ( s ) = - j -------------- ^
s + b_n-1T s
K bo
-------------------------+ . . . + bTS+bo
1
(3 .1 3 )
where, bo, b^ — , b^_^ and K are the a p p rop ria tely choosen
con sta n ts, depending upon the f i l t e r c h a r a c t e r is t ic s .
Low-
pass f i l t e r s , l i k e , Butterworth, Chebyshev, e t c . , have tr a n s fe r
fu n ction s o f th is typ e, d if f e r in g only in the ch o ice o f the
c o e ffic ie n ts .
The LC-ladder may be r e a liz e d in the OA-C form
by using id e a l flo a t in g inductance sim u lators, along with
grounded p a ssive Cs.
This i s , however, n ot a t t r a c t iv e from
The C-matching in the AP-case has been imposed by the
b a s ic stru ctu re from which the A P - f i lt e r is re a liz e d .
107
L2
Rs
L4
— •-
O+
--C 3
C,:
o --
1--------- •-
0-----------
n EVEN
n ODD
(a)
+
Cs
R2
»
—
—.
o3------- 11----'
Dl ==
R:
---- •------- -----
'n
D3:
1— 0
=C l
Dn^
^C l
------------- 9-------- •------------
1--- 0
■ ■ 0- 0— 1
n EVEN
n ODD
— o>-
(b)
Fig. 3-5 (a)
(b)
nth-order
pr o t ot yp e
FDNR-version
ladder.
LC l o w - p a s s
ladder.
of t h e p r o t o t y p e l o w - p a s s
108
the co n sid e ra tio n o f p r a c t ic a l implem entation o f the c i r c u i t .
The id e a l flo a t in g Ls req u ire a la r g e count o f a c tiv e and
p assive components, b esid es the requirement o f c r i t i c a l
matching c o n d itio n s .
T h erefore, the FDNR-based r e a liz a tio n
o f the la d d er is con sid ered .
by the complex frequency
The la d d e r network i s scaled
*s*, which transform s i t in t o the
CRD-version, shown in Fig. 3 .5 ( b ) , w ithout a ffe c t in g the
tr a n s fe r fu n ction .
The problem then reduces t o the r e a l i ­
z a tio n o f id e a l grounded FDNRs and flo a t in g Rs in the OA-C
form.
However, as i t has been p oin ted out in Chapter 2, the
id e a l sim ulators req u ire la r g e component coiant.
T herefore,
the a ctiv e-C v e rs io n o f the ladd er i s r e a liz e d in th is s e c tio n
by employing n on -id ea l immittance sim u la tors, which within
c e r ta in c o n s tr a in ts , r e a liz e t h e ir id e a l v e r s io n s .
I t may be
noted th at these c i r c u i t s were not in clu ded in Chapter 2.
Consider the n o n -id e a l grounded FDNR and the flo a t in g
R sim u lators, r e s p e c tiv e ly shown in F ig. 3 .6 (a ) and ( b ) . The
n on -id ea l grounded FDNR b a s ic a lly r e a liz e s the d r iv in g -p o in t
impedance fu n ction
s f B
1
B
Z (s ) = — p— = ----- + ^
•
s^C
sC
s^C
(3 .1 4 )
This c i r c u i t r e a liz e s a s e r ie s com bination o f an FDNR and a
c a p a c ito r .
However, in the frequency range, w << B, the
c i r c u i t r e a liz e s an e f f e c t i v e ideal-FDNR with
109
. I.
4- ^ ■_
1 o-
10 "
Z (s)-
L
'1
a
T
(a)
1o
1 o-
■ - A A /-------- -o *
T
(b)
F i g . 3*6
Tm O A - C c i r c u i t s r eal i zi ng (a) i d e a l
graunded F D N R (b) i deal f l oat i ng R ;
in the f r e q u e n c y r a n g e « « B -
■^2
110
(3 .1 5 )
S im ila rly , a n a ly sis o f the flo a tin g -R sim ulator o f
Fig. 3 .6 (b ) r e a liz e s the adm ittance-param eter m atrix :
(s-^B)
- B
( 3 . 16 )
Y = C
(s+B)
-B
In the frequency range, w << B, the m atrix reduces to
1
-1
Y = CB
(3 .1 7 )
-1
1
and r e a liz e s an e f f e c t i v e i d e a l - f l o a t i n g r e s is t o r with R = 1/CB.
These low-component sim ulators are subsequently used in
the a ctiv e-C r e a liz a t io n o f the la d d er, employing the trans­
formed c i r c u i t o f Fig. 3 .5 ( b ) .
la d d er i s given in Fig. 3 .7 .
The fin a l OA-C v e rs io n o f the
I t may be noted th at the c i r c u i t
uses a low component count o f only
c a p a cito rs f o r even n, and o f
to r s f o r odd n la d d e rs .
n OAs
and ( ^ n * 2 )
( 3 n - l ) Oas and |-(n+l) ca p a ci­
The on ly r e s t r i c t i o n w ith the c i r c u i t
i s to use i t in the lim ite d frequency range, w << B, which
r e s t r i c t s i t s use to the AF-range.
This s e c tio n demonstrates
how component saving may be achieved at the c o s t o f some
co n stra in ts imposed on the working frequency-range o f f i l t e r s .
Ill
Q
Q
O
uO
ro
<u
T3
-O
ro
if)
in
ro
a.
I
o
c
0
‘in
(U
>
O
1
<
o
CO
CD
112
5 .2
L~Based Approach to OA-C Synthesis
In the inductance (L )-b a sed approach to OA-C synthe­
s i s , the R and L elements o f the p rototyp e f i l t e r d escrib in g
the tr a n s fe r fu n ction are required t o be sim ulated in the
OA-C form and then su b stitu te d f o r the corresponding passive
components.
This r e s u lts in the OA-C v e r s io n o f the f i l t e r .
As has been seen in the preceding s e c tio n , the use o f id e a l
component sim ulators norm ally requ ires a la rg e count o f a c tiv e
and p assive components.
In a d d itio n , the use o f such compo­
nent sim u lators, p a r tic u la r ly the flo a t in g ones, may re s u lt
in u ndesirable s e n s it iv it y p r o p e r tie s and the c i r c u i t having
unaccountable p a r a s ite s , when the matching con d ition s are not
s t r i c t l y s a t i s f ie d .
A lte r n a tiv e ly , the use o f n on -id ea l
component sim ularors f o r rep la cin g a s e t o f components is
more a t t r a c t iv e from the co n sid e ra tio n s mentioned above.
In th is s e c tio n , the second approach i s th e re fo re
used in the OA-C r e a liz a tio n s o f ( i ) standard secon d -ord er
f i l t e r s and ( i i )
h ig h er-ord er la d d ers.
In S ection 3 .2 ,1 , a
standard secon d -ord er OA-C high-pass (HP) f i l t e r i s re a liz e d
by employing n on -id ea l L -sim iilator.
The technique i s then
ap plied in S ection 5 . 2 . 2 . in the r e a liz a t io n o f h ig h er-ord er
OA-C la d d ers.
113
5 .2 .1
R ea liza tion o f se con d -ord er f i l t e r with n on -ideal
L -sim ulator
This s e c tio n demonstrates the r e a liz a tio n o f a second
ord er high-pass f i l t e r from i t s p rototy p e, by using a non­
id e a l OA-C grounded L -sim u la tor.
i s shown in Fig. 3 .8 ( a ) .
RLC-tuned c i r c u i t i s
For
Its
the
The re s u ltin g c i r c u i t i s shown
a n a ly sis g iv es the tr a n s fe r fu n ction
V.
V ^ >
obtain in g i t s OA-C v e rs io n ,
d ir e c t ly sim ulated and replaced by the
OA-C sim ulator o f Fig. 2 ,3 in Fig. 3 .8 (b ).
The p a ssiv e HP-prototype
H, s^
:
N(s)
= ^
where
C
hp
C^l •
C^
o
C,
1 /2
1 2
and
C„
B,
C
Q » pS [ ^ .
'-1
“l
^2
C.
1 /2
. =2- J .
''34
(3 .1 9 )
This f i l t e r enjoys the m u ltifu n ctio n a l c a p a b ilit y and p rovides
a d d itio n a l standard secon d -ord er BP and LP responses at the
outputs o f
OA^ and 0A2 » r e s p e c t iv e ly .
The BP and LP gains
are given by :
^
(3.20)
and the other parameters o f in t e r e s t , such as, pole-w^ and
pole-U remain unaltered.
The f i l t e r parameters have the
114
■TO B E SIMULATED
1
R i
L
(a )
r REQUIRED O A - C
Co
sim u la to r
-----------
1
1
I
^^3
Vhp
_
Vbp
Vlp
__L
zr
(b)
Prototype
+
tC 2
Vi
Fig. 3-8 (a)
^
h i g h - p a s s filter.
(b) O A - C version of the prototype HPF.
115
a t t r a c t iv e fe a tu re o f being in terms o f C -r a tio s .
The s e n s i­
t i v i t y fig u r e s , f o r v a riou s f i l t e r parameters w ith resp ect to
the a c tiv e and p assive elem ents, are analysed and the fin a l
r e s iilts are in clu ded in Table 3.^ .
Expressions in the ta b le
show that the f i l t e r enjoys a t t r a c t iv e s e n s it iv it y p ro p e rtie s ;
again a l l the s e n s i t i v i t i e s being l e s s than or equal to
u nity in magnitude.
The c i r c u i t a ls o enjoys independent
e le c t r o n ic tuning o f p o le -fre q u e n cy .
However, independent
p a ssiv e co n tro l o f f i l t e r parameters i s not p resen t in th is
ca se.
In order to p rovid e dc b ia sin g and s t a b i l i t y , high­
valued r e s is t o r s are requ ired a cross the ca p a cito rs C2 and C^.
3 .2 .2
R e a liza tio n o f h ig h e r-o rd e r lad d ers
Inductance-based sim ulators o f Table 2 .1 may a ls o be
employed in the r e a liz a t io n o f v a riou s la d d e r stru ctu res in
the OA-C form, th is technique w ill p a r t ic u la r ly be su ita b le
fo r t o p o lo g ie s , which have inductance in the shunt-arms.
Two
examples are f i r s t con sid ered f o r the r e a liz a t io n o f OA-C
ladd ers having the Ls in th e shunt-arms.
'
Consider the n -p o le band-pass f i l t e r o f Fig. 3 .9 (a )
86
I t s r e a liz a t io n in the OA-C form requ ires the replacement o f
the grounded LC- arms by t h e ir sim ulated v e r s io n s .
The
grounded LC-sim ulator o f Fig. 2 .2 i s used in the r e a liz a tio n
o f a ctiv e-C v e rsio n o f the n -p o le BP f i l t e r , as shown in
Fig. 5 . 9 ( b ) .
The r e a liz e d f i l t e r only requ ires 2n OAs and
(^ n - 1 ) ca p a c ito r s .
115
H
c\j
\
H
I
CM
rH
r-H
o
vV
_v
+
CNJ
•>
CM
r—(
O o
o (—1 O u
CM
I—1
o
CM
U
rHiCvJ
(\J
rvj
N/
CM
CM
H
CJ
CM
+ O
Hu
O
^IfVJ
fT\
U
U
rHifVJ
I
I
cv
C\J
CM
r\y
—
<
H
V
(V
rH
CM
f—
1
rH o
CN
J
o ^
o o
HiCM
o o
Hicvj
I
-d- fo
o u
H|C\J
fV
CM
H
N/
CM
-d’
iH r—
(
o O -J"
o u
o u
h Ioj
HlfM
<r
o O
r-flCvl
I
I
P.
H
af
I
a
H
N/
#>
CM
CM
r-l
o
*
rH
O
u
rH
u
U
CM
cm '
rH
O
H
O
CM
u
o
4->
OQ
1s
CO
f\J
CQ
O
o
CM
U
u
o
117
— TO B E
sim u l a t e d
Cn-1
------- II— -------------------- o +
Q
{
-o-
n
(a)
F i g . 3-9 (a) Prototype n - p o l e B P - f i l t e r
(b) O A - C versfon of n - p o l e B P filter.
■o-
118
As a second example, c o n s id e r the n -th order lin e a r
phase BP f i l t e r , shown in Fig. 5 .1 0 (a )
86
.
I t s active-C
r e a liz a t io n requ ires the sim u lation and corresponding rep la­
cement o f the p assive RLC shunt-arms.
The grounded tuned
c i r c u i t sim ulator o f Fig. 2 .3 i s employed f o r the purpose.
The fin a l r e a liz a tio n i s shown in Fig. 3 .1 0 (b ), where an n -th
order f i l t e r requ ires 2n OAs and (5 n -l) ca p a cito rs in i t s
r e a liz a t io n .
The inductance-based sim u lation technique may a ls o be
employed a t t r a c t iv e ly , w ith c e r ta in c o n s tr a in ts , in r e a liz in g
s tru ctu re s, which have Ls in the flo a t in g arms, provided
m u ltip le component replacem ent i s p o s s ib le .
To demonsxrate
t h is , an n -th order Cauer-Chebyshev (CC) low -pass ladder o f
Fig. 3 .1 1 (a ) i s con sid ered .
I t s r e a liz a t io n in the Oa-C form
req u ires the sim ulation and subsequent replacement o f the
flo a t in g LC-tuned c i r c u i t .
The flo a t in g LC-sim ulator o f
Fig. 2 .2 5 (b ) already stu d ied in S e ctio n
Job.
2.5
i s used f o r the
The resu ltin g Oa -C v e r s io n o f the CC low -pass la d d er i s
shown in Fig. 3.11(t»).
2 ( n - l ) OAs and
This c i r c u i t uses a low count o f only
( 3 n - l) c a p a c it o r s f o r an n -th order la d d er.
The b a sic lim it a tio n o f the c i r c u i t i s in i t s use over a
r e s t r ic t e d frequency range o f op era tion , w << B, as discu ssed
in S ection 3 .1 .3 ^ .
Note ; In the Oa-C la d d er r e a liz a t io n s , high-valued r e s is to r s
may a p p rop ria tely be used f o r providin g dc bias to the OAs.
■ 119
TO BE s i m u l a t e d
1
I
n
(a)
Fig-3.10 (a)
Prototype
(b) O A - C
o r d e r l i near p h a s e B P f i l t e r
v e r s i o n of l i n e a r - p h a s e
B P filter.
120
L2
HI—
^
>-n-1
t^n-1
f ^3
T^n
(a)
r>
-^ < 3
=rCi
a
n -1
(b)
F i g . 3-11 (a)
(CO
order Ca a e r Chebyshev
type l o w - p a s s filter.
(b) OA — C v e r s i o n
of C C - t y p e L P filter.
n
121
3.3
FDNCAP-Based Approach t o OA-C syn th esis
In th is s e c tio n , a new approach to d ir e c t form syn­
t h e s is , c a lle d the FDNCAP-Based Approach, i s con sidered f o r
the r e a liz a tio n o f secon d -ord er and h ig h er-ord er f i l t e r s in
the OA-C form.
In th is tech n iqu e, the p a ssive FlLC-prototype
network d e scrib in g the tr a n s fe r fu n ction i s f i r s t transformed
in t o an equ ivalent network by s ca lin g each admittance o f the
o r ig in a l network (N) by the square o f the com plex-variable
( s ^ ) , as shown in Fig. 3 .1 2 .
The re s u ltin g network (N*) i s
thus obtained by con vertin g a r e s is t o r to an FDNR, an induet o r to a c a p a c ito r and a c a p a c it o r to a frequency dependent
n egative ca p a citan ce (FDNCAP) element^^.
The FDNCAP has
impedance o f the form, 1/s^ F , where F i s parameter o f the
element having the u nits o f ( farad) ^. ( ohm) ^ (f ^ . il^ ).
Thus,
the RLC-network (N) i s cen verted to a DCF-network (N^) and
sy n th esis problem becomes th a t o f the r e a liz a t io n o f FDNRs
and FDNCAPs in the OA-C form.
As w i l l be seen in the subsequent s e c t io n , the technique
w i l l prove out t o be very a t t r a c t iv e in the r e a liz a t io n o f
OA-C v e rsion s o f those s tru ctu re s which have ex cessiv e number
o f inductances in the s e r ie s , as w ell as, the shunt-arms.
In t h is method a ls o , e ith e r id e a l or n o n -id e a l FDNCAP c ir c u it s
may be used in the r e a liz a t io n o f OA-C c i r c u i t s .
However,
as has been pointed out e a r l ie r , the use o f n on -id eal FDNCa Psim ulators f o r m ultiple-com ponent replacem ent w ill prove out
122
NETWORK-N
network-N
D
o-- VV'- -- o
,V r
r
SC
V
l
VsL
J
sc
Fig. 3-12
-o i D =• V
-o J C —V
-o » F = C
V
R L C e l e m e n t s t r a n s f o r m a t i o n to D C F version by s ^ - s c a l i n g on each a d m i t t a n c e s .
123
t o be much more a t t r a c t iv e .
In S ection 3 .3 .1 , the technique
i s used, in the r e a liz a t io n o f secon d -ord er low p a s s - f i l t e r
and in S ection 3 .3 .2 f o r the r e a liz a t io n o f h igh er-ord er
la d d ers.
3 .3 .1
R ea liza tion o f secon d -ord er s e c tio n with n on -id eal
FDNCAP-simulators
In th is s e c tio n the r e a liz a t io n o f standard second-
ord er low -pass f i l t e r i s d escrib ed using an FDNCAP-based
sim u lator.
The p rototyp e secon d -ord er LP f i l t e r i s shown
in Fig. 3 .1 3 (a ).
I t s s ca lin g by s^ leads t o the FDNCAP-based
v e rs io n o f the c i r c u i t , shown in Fig. 3 .1 3 (b ).
I t may be
n o tic e d that the transformed network may com pletely be simu­
la te d by the FDNCAP-based. sim u lator o f Fig. 2.15 by taking
the responses a t the output o f Oa.^.
In the re s u ltin g c i r c u i t ,
c a p a c it o r C i s s p l i t as a p a ra lle l-co m b in a tio n o f
i.e ., C =
and C^,
fin a l OA-C r e a liz a tio n i s given in
Fig. 3 .1 3 ( c ) .
A nalysis o f the c i r c u i t y ie ld s the tra n s fe r
fu n ction :
(s ) =
V,
Vi
H,
w^
N(s)
^
D(s)
( 5. 2 1 )
where
1 /2
h p
"
% - Cb^.B2 .
.
^12
and
]
^3 ^
124
( b)
(c )
F i g . 3.13 (a) P r o t o t y p e R L C L o w _ p a s s f i l t e r ( L P F ) .
(b) F D N C A P - v e r s i o n
(c) O A - C - v e r s i o n
of L P F .
of LPF.
125
Q = [ .
®2
'^12
.
(3 .2 2 )
'^3
Once again, the f i l t e r parameters have the a tt r a c tiv e feature
o f being in terms o f C -r a tio s and the p o le -fre q u e n cy being
independently tunable w ith the b ia s -c o n t r o l.
However, inde­
pendent p assive c o n tr o l o f f i l t e r parameters i s n ot presen t.
AS the f i l t e r ’ s output is available at the output of OA, addi­
tional buffer is not needed, which is generally required in
the circu its obtained by the direct form approach.
The s e n s it iv it y fig u r e s f o r variou s f i l t e r parameters,
w ith resp ect to a c tiv e and p a ssiv e elem ents, are analysed and
th e f in a l r e s u lts are in clu ded in Table 3 .5 .
The ta b le shows
th at f i l t e r enjoys a t t r a c t iv e s e n s it iv it y p r o p e r tie s , being
le s s than or equal t o unity in magnitude.
In order t o p rovide
dc b ia sin g t o the OAs, high-valued r e s is t o r s are required
a cross
3 .3 .2
and C^.
R e a liza tio n o f h ig h e r-o rd e r ladders
Sometimes in the la d d e r r e a liz a t io n s , the L-based and
the FDNR-based techniques do not prove out to be a t t r a c t iv e ,
p a r t ic u la r ly when the stru ctu res con tain inductances in both
the s e r ie s and the shunt-arms.
In such ca s e s , the FDNCAP-
based approach g e n e ra lly lea d s t o a t t r a c t iv e r e a liz a t io n o f
the c i r c u i t in the OA-C form.
In th is s e c t io n , we s h a ll
co n sid e r two examples f o r the r e a liz a t io n o f the n -th order
126
Table 3.5 ; S e n s it iv it y fig u r e s f o r OA-C f i l t e r o f F ig .5 .1 3 (c )
127
la d d e rs , based on the FDNCAP approach.
The f i r s t example i s
based on the use o f n o n -id e a l FDNCAP f o r the sim ulation o f a
s e t o f components.
T h erefore, i t re s u lts in a low component
r e a liz a t io n w ith no matching and frequency r e s t r i c t i o n s .
In
the second r e a liz a t io n , an id e a l grounded FDNCAP i s required.
However, rath er than using an id e a l grounded FDNCAP c i r c u i t
w ith a la rg e component count and matching c o n d itio n s , the use
o f n on -id ea l FDNCAP as an id e a l one w ithin c e r ta in frequency
c o n s tr a in ts , i s suggested.
This once again r e s u lts in the
op tim iza tion o f components and elem ination o f c r i t i c a l matching
c o n d itio n s .
In the f i r s t ca se , co n s id e r the e l l i p t i c la d d er f i l t e r
o f Fig. 3 . l H a ) .
F ig. 3 . 1 ^ b ) .
I t s transform ed FDNCAP-version i s shown in
For the r e a liz a t io n o f the la d d er in the OA-C
form, two types o f n o n -id e a l grounded simulators^ I and I I ,
are requ ired.
The f i r s t type o f sim ulators ( l ) are used f o r
the replacement o f the s e r ie s C;F shunt-anns, except f o r l a s t
arm.
The FDNCAP- based sim u lator r e a liz in g grounded C:F-
fu n ction i s given in Fig. 2 .1 3 .
This c i r c u i t i s used f o r the
replacement o f the shunt-arms in the transformed c i r c u i t .
Rather than sim ulating an id e a l FDNR f o r the l a s t shunt-anm,
the com plete L -s e c tio n c o n s is tin g o f C and D i s sim ulated by
the n on -id ea l FDNR-simulator ( I I ) o f Fig. 2 .9 .
The fin a l
OA-C v ersion o f the la d d er i s shown in Fig. 2 .1 4 ( c ) .
I t is
seen th at the r e a liz a t io n does not requ ire any matching or
frequency c o n s tr a in ts , and moreover, i t uses a low count o f
128
(a)
TYPE-II
TYPE-I OA-C
Cl
sim ulato r
sim u la t o r
Cn-2
C3
Cn
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I
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____ I L
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T •s "
(b)
(c)
Fi g . 3- U (a) Prototype ell iptic l a d d e r filter.
(b) F D N C A P - v e r s i o n of e l l i p t i c l a d d e r filter.
(c) O A - C v e r s i o n of e l l i p t i c l a d d e r filter.
129
only n OAs and 2n Cs.
In the second example, the low -pass LC la d d er o f Fig.
3 .1 5 (a ) i s con sid ered .
I t s transformed FDNCAP-version i s
shown in Fig. 3 .1 5 (b ).
Now t o r e a liz e the c i r c u i t in the
OA-C form, the grounded id e a l FDNCaPs are requ ired to be
rep la ced by the corresponding OA-C s iu u la to r s .
The n on -id eal
FDNCaP o f Fig. 2.13 in the frequency range, w << B, a cts as
an id e a l grounded FDNCAP-Simulator, w ith the impedance func­
t io n , Z (s ) = l/s ^ F and F = C^/B^B^.
T herefore, f o r the r e a l i ­
z a tio n o f the la d d er in the GA-C form, t h is FDNCAP-simulator
i s used f o r the replacem ent o f a l l the shunt-arms, except
f o r the l a s t arm, which i s replaced by the FDNR-simulator
o f Fig. 3 .6 ( a ) .
Fig. 3 .1 5 ( c ) .
The OA-C v e rs io n o f the Ladder i s shown in
I t i s seen th a t the OA-C la d d er uses only
(n+1) OAs and (2 n + l) Cs f o r e v e n -n , w hile n OAs-and 2n Cs
f o r odd-n f i l t e r s .
Matching co n s tra in ts are n ot p resen t in
th is r e a liz a t io n , but frequency r e s t r i c t i o n , w << B, i s present
in the c i r c u i t .
3. ^
S3q)erimental Results
In th is s e c tio n es^Jerimental v e r i f i c a t i o n s o f the three
OA-C d ir e c t form syn th esis tech n iqu es, stu died in the preceding
s e c tio n s , have been in clu d ed .
This i s done by e:Q)erimentally
v e r ify in g , one c i r c u i t each, r e a liz e d through the d if fe r e n t
d ir e c t forro syn th esis approaches.
Since the f i l t e r s r e a liz e d
in t h is ch apter are based on the component sim ulators studied
130
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r T Y P E - II SIMULATOR
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(b )
TY P E -I O A - C SIMULATO
TYPE-ir
sim u la t o r
(C)
Fig. 3*15 (a) P rot ot y pe s i ng l y
ladder.
terminated
low-pass
(b) F D N C A P - v e r s i o n of the L P - l a d d e r .
(c) O A - C v e r s i o n of the L P - l a d d e r ,
131
in Chapter 2, the experim ental re s u lts a ls o p rovide the
v e r iX ic a tio n s on some o f the component sim ulators o f the
previous ch apter.
Lacking IG fa b r ic a tio n f a c i l i t i e s in the
Department, the experim ental v e r i f i c a t i o n s were done on the
d is c r e t e v e rsion s o f the c i r c u i t s .
The reported re s u lts have
been obtained without re s o rtin g to tuning.
3 .^ .1
Design and te s t in g o f B S - f i l t e r (FDNR-Based Approach)
The d is c r e t e v e rs io n o f the secon d -ord er band-elem ina-
tio n f i l t e r (BEF) o f Fig. 3 .3 ( c ) , r e a liz e d through the FDNRbased technique, was e 35)erim en ta lly v e r i f i e d in the la b ora tory.
I n it ia lly ^ t h e BE f i l t e r was designed f o r a ce n tre frequency
( f ^ ) o f 32.9 KHz and q u a lity f a c t o r o f 10.5 w ith OAs o f 741type and d is c r e t e Cs o f 1 p ercen t to le ra n ce .
The use o f the
design equation ( 3 . 6 ) , along w ith the measured value o f B o f
each OA as 0.9 MHz at Vc c = + 9 v o l t s , y ie ld e d the fo llo w in g
component values :
= 65 nF,
= 226 pF,
= 99.5 pF, and
= 74 nF.
The designed c i r c u i t o f Fig. 3 .1 6 (a ) was te s te d
w ith frequency response shown in Fig. 5 .1 6 (b ).
The e x p e ri­
mental r e s u lts show good con form ity with design.
The m u ltifu n ction a l c a p a b ilit y o f the f i l t e r was a ls o
v e r i f i e d experim en tally.
The previous f i l t e r , with the given
s p e c ific a t io n s and d esign , a d d itio n a lly p rovides the band­
pass c h a r a c t e r is t ic s at the output o f OA^.
The frequency
response o f th is c i r c u i t , along w ith the designed and observed
valu es o f f^ and Q, are in clu d ed in Fig. 3.17.
The low -pass
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response i s a ls o a v a ila b le a t the output term inal o f OA2 .
The f i l t e r was then s li g h t l y redesigned t o y i e l d the B utterworth response at the output o f OA2 f o r previou s p ole
frequency, f^ = 32-9 KHz.
become :
= 5 .3 nF and
remaining u naltered.
3 .1 8 (a ).
The m odified valu es o f ca p a cito rs
= 290 pF ; the oth er C-values
The low -pass f i l t e r i s shown in Fig.
I t s experim ental frequency response, along with
the th e o r e tic a l and observed valu es o f paramerers, are in c lu ­
ded in Fig. 3 .1 8 (b ).
The agreement between tn eory and
ejqjeriment i s obviou s.
The v a r ia tio n o f p o le - f ^ and p ole-Q w ith b ia s v olta g e
(V OQ) f o r the BEF o f Fig. 3 .1 6 (a ) was a ls o experim entally
v e r ifie d .
The r e s u lts are shovm in Fig. 3 .1 9 .
I t demonstrates
tne independent e le c t r o n ic tuning o f p o le -fre q u e n cy with b ia s v o lta g e over a con sid era b le range o f about 81 per ce n t, with
Q p r a c t i c a ll y remaining u naltered.
In a l l the ca se s, the experimental valu es were foxmd
to be in c lo s e con form ity w ith the th eory.
N ote, high­
valued r e s is t o r s , each o f 2.2 Mohm, were employed across
the ca p a cito rs
and C2 to p rovide the dc b ia s and s t a b i l i t y
to the OAs.
3.^ .2
Design and te s tin g o f H P -filt e r (L.-Based Approach)
The secon d -ord er high-pass f i l t e r o f Fig. 3 .3 (b ),
r e a liz e d through the L-based tech n iqu e, was designed and
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BIAS VOLTAGE (VOLTS)
Fig.3-19 Variation of p o l e - f r e q u e n c y ( U a n d p o l e - Q
w i t h b l a s - v o L t a g e (V^c) f or B E F of Fig.3-16(a).
137
experim entally v e r i f i e d by employing Bi-MOSFET input OAs,
CA 3 1 ^ .
The measured valu e o f B a t a bias o f + 9V was
found to be 4.2 MHz.
worth response a t a
The H P -filt e r was designed f o r B u tterc u t - o f f frequency, f^ = 420 KHz.
Since
the op era tion al frequency was in hundred o f k ilo h e r tz range,
i t was e s s e n tia l t o use the p r e d is to r tio n tech n iqu e, d iscu ssed
in Chapter 4.
The c i r c u i t design in corp ora tin g p r e d is to r tio n ,
along w ith ( 3 . 9 ) , y ie ld e d the fo llo w in g ca p a cito rs valu es ;
= 0.97 nF,
= 1.02 nF,
= 71.2 nF.
= ^.6 nF,
= 0.99 nF, and
This c i r c u i t i s shownin Fig. 3 .2 0 (a ).
I t was
experim entally v e r i f i e d and the frequency response i s given
in Fig. 3 . 20 ( b ) .
The r e s u lts once again e x h ib it good c o n fo r ­
mity w ith the th eory.
3 .4 .3
Desifoi and te s tin g o f L P - f i l t e r ( FDNCAP-Based Approach)
The d is c r e t e v e rs io n o f the secon d -ord er low -pass
f i l t e r o f F ig. 3 .1 3 ( c ) , r e a liz e d through the FDNCAP-based
tech n iqu e, was designed and experim en tally v e r i f i e d employ­
ing Bi-MOSFET input OAs, CA 3140.
at a bias o f + 15 V was 4.5 MHz.
The measured value o f B
The low -pass f i l t e r was
designed f o r Butterworth response a t c u t - o f f frequency, f^ =
100 KHz.
Using ( 3 .2 2 ) , the design y ie ld s the fo llo w in g capa­
c i t o r s values : C^ = 220 nF,
C^ = 100.8 nF.
= 112 pF, C^ = 3.2 nF, and
The designed c i r c u i t i s given in Fig. 3 .2 1 (a ).
The LPF was then eijqjerimentally v e r i f i e d and the frequency
response i s shown in Fig. 3 .2 1 (b ).
Once again, the re s u lts
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DE SI GNED
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(a)
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10
0-0
40
100
(b)
Fig. 3-21 (a)
(b)
De s i g n e d
200
f
(KHz)
LPF of F i g . 3-13 (c)-
Freq u en cy r e s p o n s e curve of LPF of
Fig. 3.21 (a).
lUO
were found in c lo s e con form ity with the th eory.
3.5
ConcludlnR Remarks
In th is ch ap ter, the d ir e c t form OA-C syn th esis has
been con sidered in d e t a ils .
( i ) L-Based Approach, ( i i )
(iii)
Three sjm th esis tech n iqu es, v i z . ,
FDNR-Based Approach, and ( i i i )
FDNCAP-Based Approach, have been d e scrib e d in d e t a il
and c r i t i c a l l y stu d ied f o r the r e a liz a t io n o f secon d -ord er
and h ig h er-or^ er networks in the OA-C form, s ta r tin g from the
p assive RLC- p ro to ty p e .
For the sim ulation o f components,
id e a l and n o n -id e a l immittance sim iolators, already studied
in Chapter 2, were employed.
In
a ll
the th ree approaches,
two d if fe r e n t types o f component sim ulators were requ ired.
I t has been shown that use o f id e a l sim u la tors, p a r t ic u la r ly
in the flo a tin g mode, lead s t o c i r c u i t r e a liz a t io n s with a
number o f u ndesirable fe a tu re s .
These are : ( i ) ‘ use o f la r g e
count o f a c tiv e and p a ssive components, ( i i ) requirement o f
c r i t i c a l component matching, and ( i i i ) in some c a s e s , a
r e s t r ic t e d working range o f frequency.
T herefore, in cases where id e a l component sim ulators
are e s s e n tia l, i t has been suggested to use th e corresponding
n on -id ea l sim ulators w ithin a frequency range, w << B, where
the n on -id ea l c i r c u i t s can be assumed to p rovid e the behaviour
o f id e a l sim u lators.
The use o f such n on -id ea l sim u lators,
w ith in the r e s t r ic t e d frequency range, p rovid es c i r c u i t
r e a liz a tio n s w ith low component count and w ithout requiring
141
c r i t i c a l matching c o n s tr a in ts .
An a lte r n a tiv e technique o f using n on -id ea l component
sim u la tors, without frequency r e s t r ic t io n , proves out t o be
s t i l l more a t t r a c t iv e .
In th is method, the n on -id ea l compo­
nent sim ulators used f o r the replacement o f m u ltip le-elem en ts,
e it h e r o f the p rototyp e or i t s transformed v e r s io n .
Such
r e a liz a t io n s use a low count o f a c tiv e and p a ssive components,
and do not have matching and frequency r e s t r i c t i o n s .
The s u i t a b i l i t y o f a p a r tic u la r OA-C d ir e c t form
syn th esis approach, as i s the case w ith con ven tion al active-RC
s y n th e s is , depends on the stru ctu re o f the b a sic p rototyp e
network.
For example, the L-based technique proves out t o be
a t t r a c t iv e when stru ctu res have C (s) in the flo a tin g -a r m (s )
and L or LC or RLC-network(s) in the shunt arm (s).
The FDNR
based approach, g en era lly le a d s to the r e a liz a t io n s with a
la r g e r component count.
However, the use o f n on -id ea l
component sim u la tors, w ithin the r e s t r ic t e d frequency range,
sometimes r e a liz e s f i l t e r s w ith low component count.
Struc­
tu res which req u ire, both flo a t in g and grounded in d u ctan ces,
are g e n e ra lly s u ita b le to r e a liz e w ith the d ir e c t form FDNCAP
approach.
This i s p a r tic u la r ly so i f m ultiple-com ponent
sim ulator i s employed.
The re s u ltin g OA-C networks have low
component count and a ls o do not s u ffe r from matching cons­
tr a in ts and frequency r e s t r ic t io n s .
The d iscu ssed techniques provided a u sefu l method f o r
142
the r e a liz a t io n o f b iq u a d ra tic, as w ell as, h ig h er-ord er
f i l t e r s , s u ita b le f o r m on olith ic implementation in the
MOS tech n ology.
The paramexers o f a l l the re a liz e d f i l t e r
were found to be in terms o f ra tioed -C .
c t iv e featu re f o r MOS im plem entation.
This i s very a ttra ­
Tne re a liz e d second
order f i l t e r s were g e n e ra lly found to p rov id e a d d ition a l
standard responses, besid es the one ibr wnich i t was re a liz e d .
The p o le -fre q u e n cie s o f a l l secon d -ord er f i l t e r s were found
to be tunable e l e c t r o n ic a lly w ith the b ia s -v o lta g e co n tro l
without a ffe c t in g the u o f the f i l t e r s .
A lso in many ca ses,
the c i r c u i t in corp ora tes independent p assive c o n tr o l o f pole-Q .
The s e n s it iv it y stu d ies o f the secon d -ord er f i l t e r s showed
them to have a tt r a c tiv e s e n s i t i v i t y perform ance.
F in a lly , the
ejcperimental re s u lts on the f i l t e r s , r e a liz e d by the three
syn th esis tech n iqu es, gave experim ental r e s u lts which were in
good con form ity with the th eory, w ithout re s o rtin g to tuning.