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A Universal Active R Filter
Article · July 1977
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2 authors:
Ahmed Mohamed Soliman
Mahmoud Fawzy Wagdy
Cairo University
California State University, Long Beach
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A universal active R filter
Ar.tive R filters use only resistors and op amps to realise the common transfer
functions. Ahmed M Soliman and Mahmoud Fawzy of Cairo University describe a
new active R filter.
~, R 1 /(R,+R,)
.................. (12)
From the above equations, it is seen
that an all-pass, a generalized notch and
a highpass response can be obtained as
follows:Case 1: All-pass transfer function
Equation (2) represents an all-pass
characteristics if:
. R 5 R 2 //R 3)
n = K (t.e. R = Rl- ............(13)
p
Recently, there has been a great
interest in the new class of filters, termed
active R lilter (1-3), which are based upon the one pole model of the operational amplifier.
In this paper, a new active R filter is
given. The network realizes inverting
bandpass and lowpass transfer characteristics of any arbitrary gain at two
different output terminals. At a third
output terminal, a general biquadratic
transfer characteristics is obtained,
namely, a non-minimum phase, a
generaliz€~d notch or a highpass transfer
function.
Design equations for each class of
filters are given. Sensitivities to all
active and passive circuit parameters
are shown to be very low.
.
GB 3 a(b-a-1)
p = GB 2 '
b
R,
KGB 3 .R,)
(
.
(t.e.
Ra+R. = ·as, R3
...... 14)
Case 2: Generalized notch filter
Equation (2) represents a generalized
notch filter if:
p = O(i.e. R 0 = 0, R 8 =oo) ......... (15)
All types of notch transfer characteristics
can be obtained depending on the value
ofn:
·
(i) n = K (notch filter) ............... (16)
(ii) n>K (lowpass not~) ............ (17)
(iii) n<K (highpass notch) ............ (18)
Case 3: Highpass filter
Equation (2) represents a highpass filter
if:
p = 0
................................. (I9)
n = 0 (i.e. R 5 = 0, R, =00) ......... (20)
The network
For the circuit of Fig. 1; assuming,
A1 = GBJ (i = I,2,3) ................ (1)
s
as proposed in reference (2), where GB
is the gain bandwidth product of the
operational amplifier, the transfer functions at different output terminals can
b•. calculated. By direct analysis, it is
seen that:
·
V
T 1(S) = ......:! = K
vl
..
s•+s
(wi ) +wp
R.
2
R. = D (1-K)-1 .....................(21)
where:
K = a/b ................. : ............... (3)
a = R./R 1 .... , ......................... (4)
b = I+a+R 2 /R 3 ..................... (5)
t.> 2z = m.n. GB 1 • GB~/a ............... (6)
t.> 2p = m. GB 1 • GB 2/b .................. (7)
·
Qz = "'•. a/(p.GB 2)
=!p . ym.n. a GB /GB
1
2
For equation (2), given fp, Qp, fz, Qz
and K, the design equations are:-
(2)
.........
(8)
Qp = wp. b/[GB 3 • {b-a-1)]
.
~: = ~ {t-K-~) .................. (22)
ii!
fctr;:•; . {1-K-b}-I ... (23)
=
k [~:].
~: =
n
= R,/(R,+R 6 ) ..................... (10)
m
=
GB
fcl = 2(i = 1,2,3) .................. (27)
71:
X
wP
Sx
QP
Sx
wz
s"
Qz
Sx
Rl
R2
1. a
-1. b-1
2 b
2 b
-1. a
-1. b+1
R4
R5
1. b-a-1
-b-
0
0
l.b+a+1
2
2 -.b-
0
-1
2
0
-!(1-n)
-I
1
0
2
'2
1
2
Electronic Engineering
July 1977
For a bandpass filter, given fp, Qp and
IGol, the design equations are:R. = !Go! ............................. (31)
Rt
R 1 = D-1
R•
IGol- 1 ........................ (32)
Ra=fct·fc:_,.(l-l+IGoi)-1 ... (33)
R,
f 2p
D
When IV 2 1represents a high pass filter
IVtl
(i.e. p = 0, n = 0), then:
-H,
2
-!(1-n)
2
-------
T ,(s) = -- ~ 2
V1
s +s
(wQ~P) +<op
2
(34)
For a lowpass filter given fp, Qp and
IGo j, the design equations are:
R, = !Go j .............................. (37)
R1
R,=1+1Gol ~ ....................... (38)
R3 .
D-1
Rs = fc 1 • fc 2 · [ ' -1/D
~
--y.;:- I+ IG--:f - 1.. .... (39)
J
R!!
R9
GBI GB2 GB3
-1(1-m) !(1-m)
2
2
0
0
I
2
-!(1-rn) !(1-m)
2
2
0
!(1-n)
2
-!(1-rn)
2
0
!(1-n)
-!(1-m) l.O-m)
0
2
2
!(1-m)
2
'2
0
0
1-p -(1-p)
1
I
~'
-1
1
0
2
1
2
2
I
II
r:
I
::I·•
i.i
i,i
r:
-1
2
~j
!i
h
,.
"'•l
I!
'I
0
2
2
2
I..
1\
Lowpass equations
R7
R6
J
I
I
,
which represents a lowpass filter, where:
H, = m. K. GB 1 • GB 2 ............... (35)
The lowpass gain is given by:
IGol =a= R, ........................ (36)
Rt
which can take any arbitrary value.
........................... (26)
R3
2 b
2 b
Bandpass equation_s
fc 2 Qz (
1)
Rt = fz • K • 1-K-D -1...(25)
where:
r••
n = op ·r~-
2
which represents a band pass filter, where:
H 1 = K. GB 3 ........................... (29)
The midband gain is given by:
a
R3
IGol = b-a~ = R ...............(30)
.
1
which can take any arbitrary value.
R,
v'~. b. GB 1 • GB 2 .................. ( 9)
R 7 /(Ra+R 7 ) ..................... (11)
1 s'+s ("'QpP) +wp (28)
T ,(S) = V ,/V =
"'1 ..................... (24)
=
(b-a-1). GB 3
When ~~represents a highpass filter
(i.e. p = 0, n = 0), then:
-Ht,S
v.
Design equations
s•-s ("'•) +w,•
Qz
·
.
Bandpass filter
II
I
0
49
II
____l_ -
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d
· ~·'l:.
:~
Fig. 1 .shows the circuit of the universal
active R filter. The circuit realises inverting bandpass, lowpass and general biquadratic transfer /unctions.
~
'"':.}•
·a
.I
Sensitivities
'>I
The table summarizes the sensitivities.
of wp, Qp, wz, Qz with respect to all
elements of the circuit. It is apparent
that ISwx I <0,5; LS Qx I <1
where x stands for any active or passive
circuit element, which implies very
0
small sensitivities.
j
j
';.;.>t
~~
-t
:-1
~~
:[
References
1. K. R. Rao and S. Srinivisan, "Low-Sensitivity active filters using the operational
amplifier pole," IEEE Trans. Circuits and
Systems, vol. CAS-21, pp. 260-262 March
1974.
•
2. K. R. Rao and S. Srinivisan; "A high-0
Temperature Insensitive bandpass filter using
the operational amplifier. pole", Proceadings
of the IEEE (letters), pp. 1713-1714, December 1974.
3. R. Schaum~nn, "Low~Sensitivity HighFrequency Tunable Acttve Filter without
External Capacitors," IEEE Trans. on circuits
and. systems, vol. CAS-22, No: pp. 39-44,
January 1975.
The authors are with the department of
Electronics and Communications Engineering,
Faculty of Engineering. Cairo University, Giza,
Cairo, Egypt.
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