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Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 18–21, 2009 1113
A Simple Economical Building FDNR Blocks with Modern
Operational Amplifiers
Jiřı́ Sedláček and Zoltán Szabó
BUT Brno, UTEE, Kolejnı́ 2906/4, Brno 612 00, Czech Republic
Abstract— Active RC filters designed using non-cascade filter synthesis method exhibit some
advantages — namely low sensitivities. However these parameters are paid on the other hand
due to the necessity to use more complicated building blocks which should be able to realize a
circuit simulation of required ideal RLC ladder prototype elements (e.g., ideal inductors). That
fact unfortunately brings higher filter sensitivities to real parasitic properties of active function
blocks.
Usage of new modern active elements (operational amplifiers with high GBW) and possibilities
of goal-directed lossy RLC ladder prototypes enables to design optimized ARC filter realizations
with simple and economic active building blocks. These simple grounded active selective blocks
working only with one active element enable in present time to design active filters with optimized
parameters and minimized influence of real active elements for frequency range up 1 MHz. In
paper here are some parameters of simple and economic building selective blocks with modern
active elements at higher frequencies discussed and briefly new possibilities of ARC lowpass filter
optimization in some practical examples presented.
1. INTRODUCTION
In the active filter design based on RLC ladder prototypes many different active selective building
blocks are used. By the lowpass (LP) filter design there is used very often the Bruton’s transformation, where RLC filter structure is transformed to RCD filter structure. Here are used as basic
building blocks active elements named as FDNR (frequency depended negative resistor).
In the present time a new modern active elements working successfully at higher frequency range
(voltage OAs with GBW about 1 GHz or CFA amplifiers with high frequency performance) and
using the synthesis method based on goal-lossy RLC ladder prototypes [2, 4] enable to design ARC
filters for frequency range up 1 MHz with simple and economic selective building blocks (FDNR)
working successfully only with one active element (OA).
(a)
(b)
Figure 1: Real FDNR building blocks and their transfer voltage ratio, (a) circuit with parallel lossy element,
(b) circuit with serial lossy element.
PIERS Proceedings, Moscow, Russia, August 18–21, 2009
1114
2. A SIMPLE ECONOMIC GROUNDED FDNR BUILDING BLOCKS
By LP filter design using Bruton’s transformation method there are used a real active building
blocks FDNR, which simulate impedance Z(jω), or admittance Y (jω):
1
1
− 2
jωCS
ω DS
Y(jω) = jωCP − ω 2 DP
Z(jω) =
(1)
(2)
It means that the real active block of FDNR can be modelled as serial or parallel connection of
ideal FDNR (with parameter D) and lossy kapacitor (with parameter C) how it is seen from Fig. 1.
From figure are also seen the transfer ratios of equivalent lowpass RLC and RDC circuits. The
comparison between both types of connections declare advantages of parallel circuits, the serial
circuits exhibit some decreasing of transfer slope in stop band which is depending on equivalent Q
factor value of circuit. The second effect common for both connections, which is seen from curves,
Table 1: Some practical FDNR circuit connections.
FDNR
Type
Connection
Remark
⇒
I
DS =RC
RC22
D
S
K 1
K=1
CC
C/2
S S =C/2
ω0 =
1
C RS R
Q=
1
1 RS
=
2ω 0 RC 2 R
ω0 =
II
Q=
RC2 2
DDS S RC
KK 22
1
R
= S
R
ω 0 RC
CCS S CC
C
R
DS
K
III
ω0 =
CS
C
K
Q=
DS
2
1
C RS R
RC 2
CS
1
C RS R
1
RS
=
ω 0 RC
R
C
R
⇒
C
C
K
IV
K =2
DP
DP = RC 2
CP
ω0 =
1
C RS R
Q = ω 0 RC =
CP = C
R
RS
C
R
ω0 =
K
V
DP
C
K→∞
CP
Q=
DP
RC 2
CP
2C
1
C RS R
ω 0 RC 1
=
2
2
R
RS
Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 18–21, 2009 1115
is some shift of resonant frequency of transfer function by small value of Q to smaller frequency.
This fact must be kept to filter design account in case of small value of Q factor (To the Q = 10
the resonant frequency shift is negligible, by Q = 2 is about 6%).
Some practical connections of single grounded FDNR blocks working with minimum number
of passive elements and only with one active element (e.g., OA) witch are useful by non-cascade
active filter design method were analyzed. Using circuit analysis were derived basic parameters of
all FDNR building blocks (C, D, ωw , Q) and these were ordered into Table 1 too.
Presented types of FDNR building blocks can be divided to two groups. How it is seen from
Table 1, the circuit connections labelled as FDNR I–III types exhibit equivalent serial lossy elements
(Cs ), while the types labelled as FDNR IV, V exhibit equivalent parallel lossy elements (Cp ).
3. ARC LOWPASS FILTERS DESIGNED USING ECONOMIC FDNR BUILDING
BLOCKS
By classical RLC ladder filter design there are usually single or double-sided termination RLC filter
prototypes used. In many catalogues are these standard LC ladder prototypes which are using ideal
lossless reactive L and C elements wide tabled.
Table 2: The prototypes of Tschebyschev and Butterworth single terminated RLC ladder filter (Fig. 5(b)),
normalized to ω = 1 with terminal resistor r = 1Ω — serial losses Rs .
Type
o
5
Butterworth
7
5
Tschebyshev
7
Rs
l1
c1
l2
c2
l3
c3
l4
-
1,5443
1,6936
1,3813
0,8940
0,3089
-
-
0,05
1.3487
1,7132
1.4194
0,9068
0.3355
-
-
0,10
1,1284
1,7808
1.4705
0,9247
0.3651
-
-
-
1,5571
1,7982
1,6583
1,3967
1,0546
0,6557
0,2224
0,05
1,2676
1,8322
1,6896
1,4187
1,0953
0,6709
0,2439
0,10
0,8987
2,0297
1,7600
1,4588
1,1472
0,6915
0,2685
-
2,1489
1,3016
2,6224
1,2502
1,7406
-
-
0,025
1,8980
1,2964
2,6500
1,2908
1,8893
-
-
0,050
1,5203
1,3316
2,7802
1,3656
2,0765
-
-
-
2,1828
1,3281
2,7143
1,3613
2,6752
1,2665
1,7593
0,025
1,7158
1,3066
2,7485
1,4008
2,7863
1,3409
1,9799
0,035
1,1636
1,4291
2,9934
1,4907
2,9220
1,4003
2,1028
Table 3: The prototypes of Tschebyschev and Butterworth single terminated RLC ladder filter (Fig. 5(a)),
normalized to ω = 1 with terminal resistor r = 1Ω — parallel losses Rp .
Type
o
5
Butterworth
7
5
Tschebyshev
7
Rp
l1
c1
l2
c2
l3
c3
l4
-
1,5443
1,6936
1,3813
0,8940
0,3089
-
-
6
1.1040
2.1440
1.2400
1.0290
0.3336
-
-
4
0.9630
2.3700
1.1590
1.0980
0.3450
-
-
2
0.6928
3.0570
0.9805
1.2820
0.3726
-
-
-
1,5571
1,7982
1,6583
1,3967
1,0546
0,6557
0,2224
6
0.9751
2.5700
1.2700
1.1800
0.9382
0.7628
0.2422
4
0.8167
2.9500
1.1400
2.0100
0.8876
0.8132
0.2507
2
0.5438
4.1200
0.8605
2.6100
0.7593
0.9585
0.2725
-
2,1489
1,3016
2,6224
1,2502
1,7406
-
-
6
0.8225
2.0080
2.3880
1.5428
2.6329
-
-
4
0.5053
2.8621
2.2035
1.6660
3.0700
-
-
-
2,1828
1,3281
2,7143
1,3613
2,6752
1,2665
1,7593
6
0.3768
3.5100
2.0600
2.0600
2.5300
1.4700
3.0600
4
0.2207
5.7500
1.5700
2.3800
2.2200
1.5900
3.8100
PIERS Proceedings, Moscow, Russia, August 18–21, 2009
1116
In [2] was described a new method how to optimize resulting active filter structures based on
classical ladder prototypes using goal-lossy filter prototype design, where the losses are dispersed
to the required place of RLC ladder structure.
As an example here are presented resulting normalized parameters of Tchebyshev and Butterworth types LP filters of 5th and 9th filter order with serial (Table 2) and parallel (Table 3)
lossy resistors which have been calculated using developed programs based on iterative numerical
methods.
Using this goal-directed lossy prototypes can be ARC filters (after Burton’s transformation to
RCD filter structure) above presented (parallel or serial lossy) FDNR building blocks with very
small required resulting Q factor value designed. This method of filter synthesis enables to design
ARC filters, which exhibit transfer responses very near to ideal.
4. SOME EXAMPLE OF ARC LOWPASS FILTERS DESIGNED USING ECONOMIC
FDNR BUILDING BLOCKS
As the example a lowpass Butterworth filter of 5th order with active building block FDNR V
type with parallel lossy (with cutoff frequency fc = 1 MHz, terminal resistor RL = 1000 Ω, and
operational amplifier of AD 8045 (GBW = 1 GHz) and OPA 656 (GBW = 400 MHz) has been
designed. In Fig. 2(a) is shown the basic lossy RLC prototype, in Fig. 2(b) resulting ARC structure
is seen. In Fig. 3, the calculated and measured magnitude responses of resulting filter from Fig. 2(b)
are presented. From figure is seen, that in case of higher GBW calculated and measured magnitude
responses are very closely to ideal. In case of smaller GBW the bigger slope of filter can be obtained.
R2
R3
104.7
31.2
C2
C4
220p
220p
R4
R5
13.68k
6.3k
VCC
VCC
R1
-
87
V1
(a)
C1
220p
-
IC1
+
IC2
C3
220p
C5
+
VEE
VEE
1.8n
(b)
Figure 2: LP filter of Butterworth type with FDNR V blocks (paralell losses), (a) RLC prototype structure,
(b) ARC filter structure.
dB
10
0
measured_OPA656
calculated_AD8045
-10
calculated_OPA656
measured_AD8045
-20
ideal_filt
-30
-40
-50
-60
-70
-80
-90
0,1
1
f [MHz]
10
Figure 3: Calculated and measured tranfer responses of LP Butterworth filter type of 5th order designed
using (parallel losses) FDNR V type building blocks.
Progress In Electromagnetics Research Symposium Proceedings, Moscow, Russia, August 18–21, 2009 1117
5. CONCLUSION
In this contribution are presented the simple grounded connections of selective building FDNR
blocks which enable to realize simple economic active lowpass filters. Performance of presented
FDNR blocks at higher frequencies and their parameters required to filter design is widely discussed.
One example of LP Butterworth filter type for cutoff frequency f0 = 1 MHz with parallel lossy
FDNR blocks and its performance and resulting responses with modern operational amplifiers is
here presented. Here described method of filter design based on goal-directed lossy filter RLC
prototypes enable to construct economic lowpass filters with minimum number of passive as well
of active elements. Thus the described simple economic FDNR blocks can be successfully used in
area of filter design and optimization.
ACKNOWLEDGMENT
This work has been supported by the Research Project: MSM 0021630513 and Grant Agency of
the Czech Republic under Grant 102/03/1181, 102/04/0442.
REFERENCES
1. Hájek, K. and J. Sedláček, “Kmitočtové filtry,” Vydavatelstvı́ BEN Praha, 2002.
2. Hájek, K. and J. Sedláček, “Lossy LC ladder prototypes and their use for arc filter optimization,” Wseas Transactions on Electronics, Vol. 2, No. 3, 94–99, July 2005.
3. Martinek, P. and T. Daša, “Evolutionary algoritmes by ARC filter synthesis,” ECCTD 05,
155–159, Cork, 2005.
4. Hájek, K., V. Michal, J. Sedláček, and M. Steinbauer, “A simple method of goal-directed
lossy synthesis and network optimization,” Advances in Electrical and Electronic Engineering,
249–253, Žilina, 2006.
5. Hájek, K., V. Michal, and J. Sedláček, “Modern operational amplifiers and their degeneracy
effects on active filter performance,” IC-SPETO 2006, 505–507, Gliwice-Ustroň, 2006.