47
3
ACTIVE FILTER DESIGN
WITH NOMINAL ERROR
3-0 INTRODUCTION
Although electric wave filters have been used for over a century since Marconi’s
radio experiments, the identification of stable and ideally terminated filter net-
works has occurred only during the past 35 years. Filtering at the lower instru-
mentation frequencies has always been a problem with passive filters because the
required L and C values are larger and inductor losses appreciable. The band-lim-
iting of measurement signals in instrumentation applications imposes the addition-
al concern of filter error additive to these measurement signals when accurate sig-
nal conditioning is required. Consequently, this chapter provides a development of
lowpass and bandpass filter characterizations appropriate for measurement signals,
and develops filter error analyses for the more frequently required lowpass real-
izations.
The excellent stability of active filter networks in the dc to 100 kHz instrumenta-
tion frequency range makes these circuits especially useful. When combined with
well-behaved Bessel or Butterworth filter approximations, nominal error band-
limiting functions are realizable. Filter error analysis is accordingly developed to
optimize the implementation of these filters for input signal conditioning, aliasing
prevention, and output interpolation purposes associated with data conversion sys-
tems for dc, sinusoidal, and harmonic signal types. A final section develops maxi-
mally flat bandpass filters for application in instrumentation systems.
3-1 LOWPASS INSTRUMENTATION FILTERS
Lowpass filters are frequently required to band-limit measurement signals in instru-
mentation applications to achieve a frequency-selective function of interest. The ap-
plication of an arbitrary signal set to a lowpass filter can result in a significant atten-
Multisensor Instrumentation 6
␴
Design. By Patrick H. Garrett

΃
n
+ b
n–1
΂
j
΃
n–1
+ ··· + b
0
(3-1)
f
ᎏ
f
c
f
ᎏ
f
c
48
ACTIVE FILTER DESIGN WITH NOMINAL ERROR
FIGURE 3-1. Ideal lowpass filter.
3-1 LOWPASS INSTRUMENTATION FILTERS
49
FIGURE 3-2. Butterworth lowpass amplitude.
FIGURE 3-3. Butterworth lowpass phase.
A( f ) = (3-2)
=
Bessel filters are all-pole filters, like their Butterworth counterparts. with an
amplitude response described by equations (3-3) and (3-4) and Table 3-2. Bessel

(s
ෆ
)B
ෆ
(–
ෆ
s)
ෆ
50
ACTIVE FILTER DESIGN WITH NOMINAL ERROR
FIGURE 3-4. Butterworth highpass amplitude.
TABLE 3-1. Butterworth Polynomial Coefficients
Poles nb
0
b
1
b
2
b
3
b
4
b
5
11.0
2 1.0 1.414
3 1.0 2.0 2.0
4 1.0 2.613 3.414 2.613
5 1.0 3.236 5.236 5.236 3.236
6 1.0 3.864 7.464 9.141 7.464 3.864

0
ᎏᎏ
͙
B
ෆ
(s
ෆ
)B
ෆ
(–
ෆ
s)
ෆ
3-1 LOWPASS INSTRUMENTATION FILTERS
51
TABLE 3-2. Bessel Polynomial Coefficients
Poles nb
0
b
1
b
2
b
3
b
4
b
5
11
233

FIGURE 3-7. Recommended active filter networks: (a) unity gain, (b) multiple feedback,
(c) biquad, and (d) gyrator.
S
z
Q
= ±1 passive network (3-5)
= (±1)(50 ppm/°C)(100%)
= ±0.005%Q/°C
Unity-gain networks offer excellent performance for lowpass and highpass real-
izations and may be cascaded for higher-order filters. This is perhaps the most
widely applied active filter circuit. Note that its sensitivity coefficients are less than
unity for its passive components—the sensitivity of conventional passive net-
works—and that its resistor temperature coefficients are zero. However, it is sensi-
tive to filter gain, indicating that designs that also obtain greater than unity gain
with this filter network are suboptimum. The advantage of the multiple-feedback
network is that a bandpass filter can be formed with a single operational amplifier,
although the biquad network must be used for high Q bandpass filters. However,
the stability of the biquad at higher Q values depends upon the availability of ade-
quate amplifier loop gain at the filter center frequency. Both bandpass networks can
be stagger-tuned for a maximally flat passband response when required. The princi-
ple of operation of the gyrator is that a conductance –G gyrates a capacitive current
to an effective inductive current. Frequency stability is very good, and a band-reject
filter notch depth to about –40 dB is generally available. It should be appreciated
that the principal capability of the active filter network is to synthesize a com-
plex–conjugate pole pair. This achievement, as described below, permits the real-
ization of any mathematically definable filter approximation.
Kirchoff’s current law provides that the sum of the currents into any node is
zero. A nodal analysis of the unity-gain lowpass network yields equations (3-6)
through (3-9). It includes the assumption that current in C
2

1
C
2
+
␻
C
2
(R
1
+ R
2
) + 1
V
0
ᎏ
V
i
R
2
+ 1/j
␻
C
2
ᎏᎏ
1/j
␻
C
2
V
0

– V
x
ᎏ
R
1
54
ACTIVE FILTER DESIGN WITH NOMINAL ERROR
␻
1
= and
␻
2
= and
␦
= (R
1
+ R
2
)
s
1,2
= –
␦
͙
␻
ෆ
1
␻
ෆ
2

resistor R that would provide the same average current shown by the identity of
C
2
ᎏ
2
1
ᎏ
R
2
C
2
1
ᎏ
R
1
C
1
3-2 ACTIVE FILTER NETWORKS
55
FIGURE 3-8. Unity-gain network nodal analysis.
equation (3-10). The switching rate f
s
is normally much higher than the signal fre-
quencies of interest so that the time sampling of the signal can be ignored in a
simplified analysis. Filter accuracy is primarily determined by the stability of the
frequency of f
s
and the accuracy of implementation of the monolithic MOS ca-
pacitor ratios.
R = = 1/Cf

filter implementation such as its cutoff frequency and passband flatness. Cost con-
siderations normally dictate the choice of 1% tolerance resistors and 2–5% toler-
ance capacitors. However, it is usual practice to pair larger and smaller capacitor
values to achieve required filter network values to within 1%, which results in fil-
ter parameters accurate to 1 or 2% with low tempco and retrace components.
Filter response is typically displaced inversely to passive-component tolerance,
such as lowering of cutoff frequency for component values on the high side of
their tolerance band. For more critical realizations, such as high-Q bandpass fil-
ters, some provision for adjustment provides flexibility needed for an accurate im-
plementation.
Table 3-4 provides the capacitor values in farads for unity-gain networks tabulat-
ed according to the number of filter poles. Higher-order filters are formed by a cas-
cade of the second- and third-order networks shown in Figure 3-10, each of which
is different. For example, a sixth-order filter will have six different capacitor values
and not consist of a cascade of identical two-pole or three-pole networks. Figures
3-11 and 3-12 illustrate the design procedure with 1 kHz cutoff, two-pole Butter-
worth lowpass and highpass filters including the frequency and impedance scaling
steps. The three-pole filter design procedure is identical with observation of the ap-
3-2 ACTIVE FILTER NETWORKS
57
TABLE 3-4. Unity-Gain Network Capacitor Values in Farads
Butterworth Bessel
______________________________ _____________________________
Poles C
1
C
2
C
3
C