Low Sensitivity Third Order Lowpass Butterworth Filter Using CFA

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J. of Active and Passive Electronic Devices, Vol. 5, pp. 321–326
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Low Sensitivity Third Order Lowpass
Butterworth Filter Using CFA
R. Nandi1,*, Mousiki Kar1 and Soumik Das2
Department of Electronics & Telecommunication EngineeringJadavpur
University, Kolkata-700032, India
2
Heritage Insitute of Technology, Kolkata-700107, India
1
A third−order lowpass (LP) Butterworth filter design scheme using two
CFA devices is presented. The analysis is carried out utilising the parasitic
and transadmittance poles of the AD−844 type CFA elements. The circuit
is practically active−insensitive with respect to the port mismatch errors
(∈) of the devices. The filter characteristic had been satisfactorily verified
with both PSPICE macromodel simulation and hardware circuit implementation for a frequency range 6 MHz < fo < 30 MHz.
Keywords: Butterworth filter, lowpass filter, third-order filter, CFA
1 INTRODUCTION
The current feedback amplifier (CFA) device is now well−introduced in the
literature as an active building block [1,2] that has some advantageous features over the conventional voltage operational amplifier (VOA) [3]. The significant feature of the CFA is its extended bandwidth (BW) capability that is
practically independent of the closed loop gain [4]. This makes the CFA a
superior choice over the VOA for the analog signal processing and filter
design at relatively high frequencies.
In the recent past, various CFA-based designs for first order and second order
selective circuits were proposed [5−11]; some of these schemes assumed the
parasitic/transadmittance capacitors to be very low and hence were neglected
leading to relatively low frequency designs covering upto a few hundred of KHz.
However a third−order Butterworth filter design using the CFA had not yet been
reported, albeit other designs using VOA [12] and CC II are available [13].
*Corresponding author: [email protected], [email protected]
321
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We consider here a practical CFA model [4,7,11] having parasitic elements at the non-inverting input (y) node and transadmittance components at
the current source compensation (z) node; both of these appear as grounded
shunt−RC arms. Our design approach here is to realise a third order maximally flat response LP−Butterworth filter with all the time constants normalised to equal value, while taking into account the device internal admittance
elements. In order to equalise these internal capacitors (Cy,z), we added shunt
external capacitors of appropriate value. The active sensitivity of the filter
parameters (ωo, Q) with respect to device port−tracking errors (∈) had been
shown to be extremely low. The analysis had been supported by some experimental results; the maximally flat third order LP response was verified for
the range 6 MHz < f0 < 30 MHz. with both PSPICE macromodel simulation
and hardware breadboard circuit implementation employing commercially
available CFA devices (AD-844).
2 Analysis
The proposed CFA−based circuit is shown in Fig. 1 where the device port
relations are iz = ± (1 − ∈i)ix, vx = (1 − ∈v)vy and vo = (1 − ∈o)vz. The port
mismatch errors (∈) are [10,11,14] usually extremely low |∈| < 0.03, and
they vanish for an ideal device; initially we assume ∈ = 0. For the analysis
we denote the device internal conductances by ‘g’ and external passive conductances by ‘G’.
Writing iz = ± ix for a positive/negative polarity CFA device, we get the
transfer F(s) = Eo/Ei in the form
F(s) =
Ei
K
d 3s + d 2 s2 + d1s + 1
(1)
3
x
R1
vo
−CFA
y
R2
y
z
Eo
+CFA
x
z
C1
R0
C3
R3
C2
Figure 1
CFA-based third-order filter.
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Low Sensitivity Third Order Lowpass Butterworth Filter Using CFA
323
where
K = Go/Gz2t
d1  = [{b Cyt + (1 + n) Cz1t }/G1 }] + (Cz2t/Gz2t)
(2)
d2 = [Cz1t Cyt + p Cz2t { b Cyt + (1 + n) Cz1t}]/G1 G2
(3)
d3 = Cz1t Cz2t Cyt/G1 G2 Gz2t
(4)
b = gz1/G2; n = gt/G1; p = G2/Gz2t
(5)
Cz1t = C1 + Cz1; Cz2t = Cz2 + C3; Cyt = Cy1 + Cy2 + C2
(6)
and
Gz2t  = gz2 + G3; gt = gy1 + gy2
(7)
Here ry,z (=1/gy,z) and Cy,z denote the device internal elements; as per data
sheet [16] these values are Cy ≈ 3.5 pF, ry ≈ 3 M.ohm and Cz ≈ 6 pF, rz ≈ 5
M.ohm. Since we intended to design the filter at MHz ranges, R1,2,3 (=1/
G1,2,3) were selected in K.ohm ranges, which implies b, n << 1 and, p ≈ G2/
G3. Also we equalised Cz1t = Cz2t = Cyt = C by taking C1 = C - Cz1 , C2 =
C − (Cy1 + Cy2) and C3 = C − Cz2. For a typical value of C = 8 pF (say),
we set C1,3 = 2 pF and C2 = 1 pF. Further one may consider a simplified
design with R1,2,3 = R. With these design approach, the denominator coefficients reduce to d1 = d2 = 2, and assuming the time constant is normalised to unity [12], we get
F(s) = 1/{s3 + 2s2 + 2s + 1}
(8)
which is a third order LP Butterworth filter function.
3 Sensitivity
Re−analysis of Fig. 1 with finite tracking errors (∈ ≠ 0) yields the modified
coefficients as
đ3 = (1 + ∈t1); đ2 = 2 (1 + ∈t1); đ1 = (2 + ∈t1)
(9)
Ќ = (1 + ∈t1 - ∈t2)
(10)
where ∈t1,2 = ∈i1,2 + ∈v1,2 + ∈o1,2; product of errors are neglected since
|∈| << 1.
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The sensitivities of the filter parameter are derived using Smu  = (∂m/m)/
(∂u/u), and considering [17], we may write
Sωu o = Sdu1 − Sdu2 and
SQu = Sdu3 − Sdu2 − Sdu1 The sensitivity figures are listed below in Table-1 where µ = (1 + ∈t1). It may
be seen that the proposed circuit is practically active-insensitive wherein the
sensitivities,in general, may be summarised as
− (S∈i,v.o 1)Q ≈ ∈i,v.o 1 << 1
(S∈i,v.o 2 )Q = 0 = (S∈i,v.o 1,2)ωo
4 Experimental Result
The proposed circuit had been simulated with PSPICE macromodel data of
the AD−844 device. The negative polarity device had been implemented by
cascading a high output impedance current mirror [14] with the CFA. The
components selected were R = 800 Ω and C = 8 pF ; the CFAs were supplied
with 0 + 9 V regulated d.c. bias. A satisfactory maximally−flat LP Butterworth filter response at fo = 25.3 MHz as shown in Fig. 2 had been obtained.
5 Conclusion
A new CFA−based circuit for realising third−order lowpass Butterworth filter
is presented. The analysis is carried out utilising the parasitic capacitors of
Table 1
Sensitivity characteristics of the Butterworth filter.
m
d1
d2
d3
ωo
Q
∈i1
∈i1/µ
∈i1/µ
∈i1/µ
0
− ∈i1/µ
∈v1
∈v1/µ
∈v1/µ
∈v1/µ
0
− ∈v1/µ
∈o1
∈o1/µ
∈o1/µ
∈o1/µ
0
− ∈o1/µ
u
∈i2
0
0
0
0
0
∈v2
0
0
0
0
0
∈ o2
0
0
0
0
0
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Low Sensitivity Third Order Lowpass Butterworth Filter Using CFA
325
0
-5
-10
dB -15
-20
-25
-30
1
10
100
MHz
Figure 2
Experimental results on maximally flat third-order filter response at fo = 25.3 MHz
using C (equalized value) = 8 pF; C1, 3 = 2 pF;
C2 = 1 pF; R1,2,3 = 800Ω.
by simulation
: ______________
by hardware implementation : ______________
the high−frequency model of the AD−844 CFA device; the passive−RC components of the circuit had then been approximately chosen to obtain a normalised filter function [12]. The effect of rx (≈ 40Ω) at the x−node of the device
had been neglected since rx can be virtually eleminated in some improved [4]
CFA−models. The proposed topology would be suitable for IC adaptation
since all capacitors are grounded and the circuit is practically active−insensitive.
PSPICE macromodel simulation as well as hardware circuit tests verified satisfactorily the maximally−flat third-order lowpass Butterworth response for a
range of select frequencies 6 MHz < fo < 30 MHz.
It may be mentioned in conclusion that improved versions of the CFA
are now being reported in the literature [18,19]; some of these devices,
viz, 0PA-695 IC, provide wide bandwidth (∼ 450 MHz ) at very high slewrate (∼ 4.3 KV/µ.sec). Use of such a device would enhance the operating
range of the proposed Butterworth filter with a practically active-insensitive feature.
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