Design of a Combined Filter to Reduce the Attenuation Decline in
Magnitude Response
BOHUMIL BRTNÍK
Faculty of Electrical Engineering and Informatics
University of Pardubice
Náměstí Čs. Legií 232, 530 02 Pardubice
CZECH REPUBLIC
Abstract: - Certain biquad filters structures are characterized by a reduction of the attenuation at higher
frequencies, caused by the finite value of the output resistance of the operational amplifier. In this paper, the
design of the combined BP filter without the described decreasing is discussed for which one possible solution
is to replace one of the biquads with another biquad that does not have this property. The result is a combined
filter from different structures. The methodology to design a combined BP filter without the described
decreasing is described in detail with the parameters of such biquads. The results of the proposed numerical
procedure are verified by computer simulation, for this purpose a SPICE-like program MC-10 is used.
Key-Words: - Biquad, band pass filter, structure SK, structure H, higher order filter, magnitude response
Received: October 26, 2022. Revised: March 13, 2023. Accepted: April 6, 2023. Published: May 4, 2023.
1 Introduction
The band-pass active RC (ARC) filters of certain
structures, for example, the multi-feedback
Huelsman structure (H), [1], [2], [3], [4], and more
others, are based on the biquad polynomial. the
structure’s frequency response, namely the decline
in the attenuation at higher frequencies is shown in
Figure 1. After its transient frequency, for the
operational amplifier can only be considered two
main circuit elements: input Ri and output Ro
resistance alternatively, input gi and output go
admittance. Generally, this challenge also exists in
relation to the band-pass type Delyiannis filter, [4].
Fig. 1: The decline in the attenuation in magnitude
response for the BP filter
Figure 2 depicts the cause for the band pass
(BP) filter. Specifically, the top part of Figure 2
illustrates a band pass ARC biquad multi-feedback
structure H. At the highest frequencies, all
capacitors behave like a short circuit and the
operational amplifier loses the open loop gain A, [5].
The reason for the above described decline is a
capacitor C. For that zero impedance connects to the
highest frequency when the amplification factor of
operational amplifier A is already reduced to zero,
for both input and output, as shown in the equivalent
circuit in Figure 2.
Fig. 2: Capacitors C zero impedance is the reason
for the decline of the magnitude characteristic
In this case, the circuit is only a voltage divider.
However, the magnitude of the voltage transfer ratio
at the highest frequencies must be not equal to zero.
Thus, this characteristic deviates from the
characteristic of an ideal filter.
ideal OpAmp
real OpAmp
AV
[dB] f [Hz]
fC1 fC2 fT
_
+
R
C
G
C
gigo
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2 Problem Solution
A proposed solution to the above described problem
is a combined filter, which can be designed in
several ways. For example, one potential solution is
the recalculation of part of the filter (i.e.
recalculation of the circuit parameters of one biquad
of the same structure to another biquad structure). A
second approach is to combine filter design, which
consists of several filter structures (biquads) in
partial filter part (biquad) direct design (i.e. without
recalculation of the circuit parameters). This article
describes a novel design procedure in Section 3.
3 Combined Filter Design
This section describes the design procedure for the
BP combined filter. The BP filter can be
implemented as a BP filter and/or as an HP+LP
filter, the selection criteria are formalized in (1), [3],
[4].
LPHP2
BP2
f
f
MIN
MAX
(1)
However, In the case of an even-order biquad of
narrow-band band-pass filters, BP-H exhibits a
decreased attenuation of the transition frequency of
the operational amplifier at high frequencies, i.e.
frequencies in the leaky proof zone, which again
degenerates the properties of this filter. This
decrease in attenuation of transit frequency exhibits
not only the three well-known structures of BP-H,
[6], [7], two of which are shown by Figure 3.
Fig. 3: The other two BP-H filter variants.
Consider an active RC band-pass filter with an
operational amplifier and Butterworth
approximation function for the center passband
frequency fC = 10.103 Hz, for the passband Δf = 103
Hz the attenuation is AC = 3 dB. For stopband B =
4.103 Hz, the stopband attenuation is AS = 20 dB.
The filter calculation steps are as follows, [8], [9].
1) Described filter specification: is shown in Figure
4.
Fig. 4: Filter specification
2) Both stop frequencies are calculated as:
(2)
Hz8198
Hz12198
2
10.4
)
2
104
()1010(
3
2
3
23
3) The following are normalized stop frequencies:
004
1012198
1000012198
fΔf
ff
F3
22
1S
2
0
2
1S
1S ,
(3)
004
108198
100008198
fΔf
ff
F3
22
2S
2
0
2
2S
2S ,
(4)
4) Therefore the Coefficient for stop frequency is:
004004004FFk 2S1S ,),;,min();min(
(5)
3) Coefficient of damping: d is given by
100
110
110
110
110
d310
2010
A10
A10
C
S
,
,
,
,
(6)
4) Order of filter: n is given by (7)
661
42
100
k2
d
n,
log
log
log
log
(7)
Thus must be:
2n
(8)
5) Coefficient of the wide band: k is given by (9):
2
fΔ
)
2
fΔ
(ff 22
02,1C
(9)
Hz9512
Hz10512
2
10
)
2
10
()10.10(
3
2
3
23
thus:
21051
9512
10512
f
f
k
1C
2C ,
(10)
5) The Butterworth approximation coefficients,
[10], [11], are presented in Table 1.
_
+
_
+
fC2
AC
AS
fS2
A[dB]
f[Hz]
fC1
fS1 fO
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Table 1. Butterworth Approximation Coefficients
6) Coefficient
α
is given by (11):
20
102
10102
fΔF
f2
α3
3
0
0
(11)
7) Quality factor of sections
Q
~
:
2222 )
Q
~
α
()α1(α1
2
Q
~
Q
(12)
2222 )
7071,0
20
()201(201
2
7071,0
1514,
8) Coefficient
F
k
:
1)
Q
~
α
Q
(
Q
~
α
Q
k2
F
(13)
036,11)
7071,020
15,14
(
7071,020
15,14 2
9) The passband gain of the partial filters
1)
k
1
k(QA
F
F0
(14)
416,11)
036,1
1
036,1(15,14
10) The middle frequencies for partial filters are
Hz9653
0361
10
k
f
f
4
F
0
01 ,
(15)
Hz10360100361kff 4
F002 ,
(16)
11) Capacity is determined by (17)
F10
10
10
f
10
C9
4
7
C
7
(17)
and/or can be chosen, for example as
F1010C9
(18)
12) Resistors must be calculated in following order
R13, R11, R12, the value for 1st section are:
Ω107046
10109653π
1514
Cfπ
Q
R3
9
01
13
,
,
(19)
001
11 ACfπ2
Q
R
(20)
Ω1050,16
1416,110109653π2
15,14 3
9
Cfπ)AQ2(
Q
R
010
2
12
(21)
Ω1,58
10109653π)416,115,142(
15,14
92
and for 2nd section:
Ω105043
101010360π
1514
Cfπ
Q
R3
9
02
23
,
,
(22)
1416,1101010360π2
15,14
ACfπ2
Q
R9
002
21
Ω1040,15 3
(23)
Cfπ)AQ2(
Q
R
020
2
22
(24)
Ω2,54
101010360π)416,115,142(
15,14
92
13) Circuit diagram of the designed filter is in
Figure 5.
Fig. 5: Designed circuit
14) It is required to use an operational amplifier,
whose transient frequency fT is given as (25):
Qf200fOT
(25)
where Q is the quality. Thus must be:
Hz108,215,141010200Qf200f73
OT
(26)
That means it is necessary to choose
MHz28fT
(27)
Therefore an operational amplifier LF 400 C is used
for example.
15) Simulation of calculated results.
First, the spice-like program MC-10 was used for
the simulation of calculated results. The real
operational amplifier LF 400 C is used with the
following parameters Ao=200 K, GBW = 18 MEG,
and ROUT = 50 Ω. The simulation of the calculated
results by the MC-10 program is shown in Figure 6.
15k4 10k
10k
54
43k5
-
+
C21
R21
R22
R23
C22
16k56 10k
10k
58
46k7
-
+
C11
R11
R13
R12
C12
N
FO
Q
2
1
0,7071
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Fig. 6: Simulation results if Op.Amp. LF400C,
GBW 18 MHz is used.
In another case, if the equation
Qf200fOT
is
not considered for the design, then there will be a
frequency shift. In this case, the simulation of the
results of Op. Amp with GBW 1 MHz is used are
depicted in Figure 7, whereas, the detail around the
center frequency of 10 kHz is in Figure 8.
Fig. 7: Frequency shift if Op.amp.: LM 741, GBW 1
MHz is used
Fig. 8: Frequency shift in detail from Figure 7 if
Op.Amp. LM741, GBW 1 MHz is used.
16) Correction of the attenuation decline at high
frequencies.
As we can see from Figure 6, the attenuation
does not decrease at high frequencies, it stabilizes at
a constant value. This disadvantage has not on the
contrary BP-SK with a grounded capacitor at the
input. Its inclusion in the cascade can then suppress
this disadvantage.
17) Calculation of SK-BP biquad
The first section with BP-H will be changed by the
BP-SK filter.
Capacity is determined by (28)
F10
10
10
f
10
C9
4
7
C
7
(28)
and/or can be chosen, for example as
F1010C9
.
Fig. 9: Circuit diagram of combined recalculated SK-H BP filter.
100
1K
10K
100K
1M
10M
100M
-125.00
-100.00
-75.00
-50.00
-25.00
0.00
25.00
dB(v(11))
F (Hz)
Micro-Cap 10 Evaluation Version
circuit_2xBP_2or_designed-GBW18.cir
100
1K
10K
100K
1M
10M
100M
-125.00
-100.00
-75.00
-50.00
-25.00
0.00
25.00
dB(v(11))
F (Hz)
Micro-Cap 10 Evaluation Version
circuit_2xBP_2or_designed-GBW1.cir
5K
10K
20K
-50.00
-25.00
5.00
dB(v(11))
F (Hz)
Micro-Cap 10 Evaluation Version
circuit_2xBP_2or_designed-GBW1.cir
15k4 10k
10k
54
43k5
-
+
C22
R22
R21
R23
C21
10k
1k929
1k
1k648
3k296
-
+
R12
R2
R1
C12
10k
R11
1k648
C11
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18) The resistor R is given by
Ω1648
10109653π2
1
Cfπ2
1
R9
01
(29)
19) Resistors for amplifier feedback to amplify
adjust are given by:
Ω1929
1514
115142
Q
1Q2
R2
,
,
(30)
Ω1000
15,14
1
2
1929
Q
1
2
R
R2
1
(31)
20) Circuit diagram of combined SK-H BP filter
The circuit diagram of the combined SK-H BP filter
is shown in Figure 9.
4 Simulation Results
The results of the proposed numerical procedure are
verified by computer simulation, [12], [13]. Next for
the simulation of calculated results the spice-like
program MC-10 was used. The operational
amplifier is used LF 400 C with parameters Ao=200
K, GBW = 18 MEG, ROUT = 50 Ω, i.e. it is the real
operational amplifier. Simulation of calculated
results by the spice-like program MC-10 is shown in
Figure 10.
Fig. 10: Circuit magnitude characteristic after
recalculation to a combined filter, Op. Amp. LF
400C, GBW 18 MHz is used
Figure 11 depicts the detail around the center
frequency of 10 kHz.
Fig. 11: Circuit magnitude characteristic from
Figure 10 in detail, Op. Amp. LF 400C, GBW 18
MHz is used
5 Discussion
The magnitude and phase characteristics before and
after recalculation are shown in Figure 6 and Figure
10. As is depicted, the phase characteristics in
Figure 12 and Figure 13 are identical, only the phase
is decreased by 180° due to connection of a
noninverting operational amplifier in the H-filter
structure, as we can see. The phase is increased by
180° due to the use of a noninverting operational
amplifier connection. If this is a problem, it can be
easily eliminated by including another inverting
amplifier as a voltage follower in the cascade.
Fig. 12: Circuit phase characteristic
Fig. 13: Circuit phase characteristic after
recalculation to a combined filter.
Another design procedure with recalculation of one
biquad is described in [14], for the LP filter.
6 Conclusion
In many cases, the Huelsmann structure is a very
popular filter structure due to the achievement of
high-quality Q. The disadvantage of the band-pass
filter of the Huelsmann structure, i.e. the decrease of
the attenuation above the transient frequency of the
operational amplifier, is discussed. Here, A new
filter structure is proposed to eliminate this
disadvantage. In the first step, the general filter
structure must be designed. If the filter structure will
be chosen first, the design steps 19, 20, and 21 can
be omitted. The results of the simulations show a
monotonic decrease in the amplitude characteristic,
without a decrease in the attenuation.
100
1K
10K
100K
1M
10M
100M
-200.00
-175.00
-150.00
-125.00
-100.00
-75.00
-50.00
-25.00
0.00
25.00
dB(v(15))
F (Hz)
Micro-Cap 10 Evaluation Version
circuitSK_H_BP_2or_GBW18.cir
5K
10K
20K
-50.00
-25.00
5.00
dB(v(15))
F (Hz)
Micro-Cap 10 Evaluation Version
circuitSK_H_BP_2or_GBW18.cir
100
1K
10K
100K
1M
10M
100M
-400.00
-240.00
-120.00
0.00
120.00
200.00
ph(v(11)) (Degrees)
F (Hz)
Micro-Cap 10 Evaluation Version
circuit_2xBP_2or_designed-GBW1.cir
100
1K
10K
100K
1M
10M
100M
-600.00
-496.00
-372.00
-248.00
-124.00
20.00
ph(v(15)) (Degrees)
F (Hz)
Micro-Cap 10 Evaluation Version
circuitSK_H_BP_2or_GBW18.cir
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This new design combined filter procedure is
described for band-pass structure only, but it can be
used for other filter structures as well.
Based on up-to-date published research projects,
a filter assembled from biquads of the same
structure was designed, and then one biquad was
selected and recalculated to another more suitable
type. The contribution of this research work is the
proposal of a procedure for the direct design of a
combined filter, when biquads of different structures
are directly designed, i.e. without the need for
subsequent conversion of one type of biquad to
another type.
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[12] M. Štork, D. Mayer, J. Hrušák, One nonlinear
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[14] Brtnik, B, Design a Combination of Filters to
Reduce Decline the Attenuation in Magnitude
Response. WSEAS Transaction on Circuits
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Contribution of Individual Authors to the
Creation of a Scientific Article (Ghostwriting
Policy)
The author contributed in the present research, at all
stages from the formulation of the problem to the
final findings and solution.
Sources of Funding for Research Presented in a
Scientific Article or Scientific Article Itself
No funding was received for conducting this study.
Conflict of Interest
The author has no conflict of interest to declare.
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(Attribution 4.0 International, CC BY 4.0)
This article is published under the terms of the
Creative Commons Attribution License 4.0
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