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Active Filter Design Techniques
Thomas Kugelstadt
16.1 Introduction
What is a filter?
A filter is a device that passes electric signals at certain frequencies or frequency ranges while preventing the passage of others. — Webster.
Filter circuits are used in a wide variety of applications. In the field of telecommunication, band-pass filters are used in the audio frequency range (0 kHz to 20 kHz) for modems and speech processing. High-frequency band-pass filters (several hundred MHz) are used for channel selection in telephone central offices. Data acquisition systems usually require anti-aliasing low-pass filters as well as low-pass noise filters in their preceding sig- nal conditioning stages. System power supplies often use band-rejection filters to sup- press the 60-Hz line frequency and high frequency transients.
In addition, there are filters that do not filter any frequencies of a complex input signal, but just add a linear phase shift to each frequency component, thus contributing to a constant time delay. These are called all-pass filters.
At high frequencies (> 1 MHz), all of these filters usually consist of passive components such as inductors (L), resistors (R), and capacitors (C). They are then called LRC filters.
In the lower frequency range (1 Hz to 1 MHz), however, the inductor value becomes very large and the inductor itself gets quite bulky, making economical production difficult.
In these cases, active filters become important. Active filters are circuits that use an op- erational amplifier (op amp) as the active device in combination with some resistors and capacitors to provide an LRC-like filter performance at low frequencies (Figure 16–1).
L R
C
VIN VOUT VIN
VOUT
R 1
C 1
C 2
R 2
Figure 16–1. Second-Order Passive Low-Pass and Second-Order Active Low-Pass
Chapter 16
Fundamentals of Low-Pass Filters
This chapter covers active filters. It introduces the three main filter optimizations (Butter- worth, Tschebyscheff, and Bessel), followed by five sections describing the most common active filter applications: low-pass, high-pass, band-pass, band-rejection, and all-pass fil- ters. Rather than resembling just another filter book, the individual filter sections are writ- ten in a cookbook style, thus avoiding tedious mathematical derivations. Each section starts with the general transfer function of a filter, followed by the design equations to cal- culate the individual circuit components. The chapter closes with a section on practical design hints for single-supply filter designs.
16.2 Fundamentals of Low-Pass Filters
The most simple low-pass filter is the passive RC low-pass network shown in Figure 16–2.
R
C
VIN VOUT
Figure 16–2. First-Order Passive RC Low-Pass
Its transfer function is:
A(s)
RC
s 1 RC
1 1 sRC
where the complex frequency variable, s = j ω + σ , allows for any time variable signals. For pure sine waves, the damping constant, σ, becomes zero and s = j ω.
For a normalized presentation of the transfer function, s is referred to the filter’s corner frequency, or –3 dB frequency, ωC, and has these relationships:
s
s C
j
C
j
f fC
j!
With the corner frequency of the low-pass in Figure 16–2 being fC = 1/2 π RC , s becomes s = sRC and the transfer function A(s) results in:
A(s)
1 1 s
The magnitude of the gain response is:
|A|
1
1 !^2
For frequencies Ω >> 1, the rolloff is 20 dB/decade. For a steeper rolloff, n filter stages can be connected in series as shown in Figure 16–3. To avoid loading effects, op amps, operating as impedance converters, separate the individual filter stages.
Fundamentals of Low-Pass Filters
Frequency — Ω
Ideal 4th Order Lowpass
4th Order Lowpass
1st Order Lowpass
|A| — Gain — dB
Frequency — Ω
Ideal 4th Order Lowpass
4th Order Lowpass
1st Order Lowpass
φ^
— Phase — degrees
Note: Curve 1: 1st-order partial low-pass filter, Curve 2: 4th-order overall low-pass filter, Curve 3: Ideal 4th-order low-pass filter
Figure 16–4. Frequency and Phase Responses of a Fourth-Order Passive RC Low-Pass Filter
The corner frequency of the overall filter is reduced by a factor of α ≈ 2.3 times versus the –3 dB frequency of partial filter stages.
Fundamentals of Low-Pass Filters
Active Filter Design Techniques 16-
In addition, Figure 16–4 shows the transfer function of an ideal fourth-order low-pass func- tion (Curve 3).
In comparison to the ideal low-pass, the RC low-pass lacks in the following characteris- tics:
The passband gain varies long before the corner frequency, fC, thus amplifying the upper passband frequencies less than the lower passband.
The transition from the passband into the stopband is not sharp, but happens gradually, moving the actual 80-dB roll off by 1.5 octaves above fC.
The phase response is not linear, thus increasing the amount of signal distortion significantly.
The gain and phase response of a low-pass filter can be optimized to satisfy one of the following three criteria:
A maximum passband flatness,
An immediate passband-to-stopband transition,
A linear phase response.
For that purpose, the transfer function must allow for complex poles and needs to be of the following type:
A(s)
A 0
! 1 a 1 s^ ^ b 1 s
(^2) "! 1 a 2 s^ ^ b 2 s
(^2) " (^) ###! 1 a ns^ ^ bns
(^2) "
A 0
i
! 1 a is^ ^ bis
(^2) "
where A 0 is the passband gain at dc, and ai and bi are the filter coefficients.
Since the denominator is a product of quadratic terms, the transfer function represents a series of cascaded second-order low-pass stages, with ai and bi being positive real coef- ficients. These coefficients define the complex pole locations for each second-order filter stage, thus determining the behavior of its transfer function.
The following three types of predetermined filter coefficients are available listed in table format in Section 16.9:
The Butterworth coefficients, optimizing the passband for maximum flatness
The Tschebyscheff coefficients, sharpening the transition from passband into the stopband
The Bessel coefficients, linearizing the phase response up to fC
The transfer function of a passive RC filter does not allow further optimization, due to the lack of complex poles. The only possibility to produce conjugate complex poles using pas-
Fundamentals of Low-Pass Filters
Active Filter Design Techniques 16-
16.2.2 Tschebyscheff Low-Pass Filters
The Tschebyscheff low-pass filters provide an even higher gain rolloff above fC. However, as Figure 16–6 shows, the passband gain is not monotone, but contains ripples of constant magnitude instead. For a given filter order, the higher the passband ripples, the higher the filter’s rolloff.
Frequency — Ω
2nd Order
4th Order
9th Order
|A| — Gain — dB
Figure 16–6. Gain Responses of Tschebyscheff Low-Pass Filters
With increasing filter order, the influence of the ripple magnitude on the filter rolloff dimin- ishes.
Each ripple accounts for one second-order filter stage. Filters with even order numbers generate ripples above the 0-dB line, while filters with odd order numbers create ripples below 0 dB.
Tschebyscheff filters are often used in filter banks, where the frequency content of a signal is of more importance than a constant amplification.
16.2.3 Bessel Low-Pass Filters
The Bessel low-pass filters have a linear phase response (Figure 16–7) over a wide fre- quency range, which results in a constant group delay (Figure 16–8) in that frequency range. Bessel low-pass filters, therefore, provide an optimum square-wave transmission behavior. However, the passband gain of a Bessel low-pass filter is not as flat as that of the Butterworth low-pass, and the transition from passband to stopband is by far not as sharp as that of a Tschebyscheff low-pass filter (Figure 16–9).
Fundamentals of Low-Pass Filters
Frequency — Ω
Butterworth
Bessel
Tschebyscheff
φ
— Phase — degrees
Figure 16–7. Comparison of Phase Responses of Fourth-Order Low-Pass Filters
Frequency — Ω
Butterworth Bessel
Tschebyscheff
T
gr
— Normalized Group Delay — s/s
Figure 16–8. Comparison of Normalized Group Delay (Tgr) of Fourth-Order Low-Pass Filters
Fundamentals of Low-Pass Filters
Frequency — Ω
1st Stage
2nd Stage
3rd Stage
4th Stage 5th Stage
Overall Filter Q^5
|A| — Gain — dB
Figure 16–10. Graphical Presentation of Quality Factor Q on a Tenth-Order Tschebyscheff Low-Pass Filter with 3-dB Passband Ripple
The gain response of the fifth filter stage peaks at 31 dB, which is the logarithmic value of Q 5 :
Q 5 [dB] 20·logQ 5
Solving for the numerical value of Q5 yields:
Q 5 10
31 (^20) 35.
which is within 1% of the theoretical value of Q = 35.85 given in Section 16.9, Table 16–9, last row.
The graphical approximation is good for Q > 3. For lower Qs, the graphical values differ from the theoretical value significantly. However, only higher Qs are of concern, since the higher the Q is, the more a filter inclines to instability.
16.2.5 Summary
The general transfer function of a low-pass filter is :
A(s) (16–1)
A 0
i
! 1 a is^ ^ bis
(^2) "
The filter coefficients ai and bi distinguish between Butterworth, Tschebyscheff, and Bes- sel filters. The coefficients for all three types of filters are tabulated down to the tenth order in Section 16.9, Tables 16–4 through 16–10.
Low-Pass Filter Design
Active Filter Design Techniques 16-
The multiplication of the denominator terms with each other yields an n th^ order polynomial of S, with n being the filter order.
While n determines the gain rolloff above fC with $^ n·20 dB%decade, ai and bi determine the gain behavior in the passband.
In addition, the ratio
bi ai^ ^ Q
is defined as the pole quality. The higher the Q value, the
more a filter inclines to instability.
16.3 Low-Pass Filter Design
Equation 16–1 represents a cascade of second-order low-pass filters. The transfer func- tion of a single stage is:
A (16–2) i(s)^ ^
A 0
! 1 a is^ ^ bis
(^2) "
For a first-order filter, the coefficient b is always zero (b 1 =0), thus yielding:
A(s) (16–3)
A 0
1 a 1 s
The first-order and second-order filter stages are the building blocks for higher-order fil- ters.
Often the filters operate at unity gain (A 0 =1) to lessen the stringent demands on the op amp’s open-loop gain.
Figure 16–11 shows the cascading of filter stages up to the sixth order. A filter with an even order number consists of second-order stages only, while filters with an odd order number include an additional first-order stage at the beginning.
Low-Pass Filter Design
Active Filter Design Techniques 16-
R 1 R
2
C 1
VIN
VOUT
Figure 16–13. First-Order Inverting Low-Pass Filter
The transfer functions of the circuits are:
A(s)
1
R 2
R 3
1 (^) cR 1 C 1 s
A(s)
$
R 2
R 1
1 (^) cR 2 C 1 s
and
The negative sign indicates that the inverting amplifier generates a 180° phase shift from the filter input to the output.
The coefficient comparison between the two transfer functions and Equation 16–3 yields:
A 0 1
R 2
R 3
A 0 $
R 2
R 1
and
a 1 (^) cR 1 C 1 a 1 ^ cR 2 C 1
and
To dimension the circuit, specify the corner frequency (fC), the dc gain (A 0 ), and capacitor C 1 , and then solve for resistors R 1 and R 2 :
R 1
a 1
2 $fcC 1
R 2
a 1
2 $fcC 1
and
R 2 R 3 !A 0 $ 1 "^ R 1 $^
R 2
A 0
and
The coefficient a 1 is taken from one of the coefficient tables, Tables 16–4 through 16– in Section 16.9.
Note, that all filter types are identical in their first order and a 1 = 1. For higher filter orders, however, a 1 ≠1 because the corner frequency of the first-order stage is different from the corner frequency of the overall filter.
Low-Pass Filter Design
Example 16–1. First-Order Unity-Gain Low-Pass Filter
For a first-order unity-gain low-pass filter with fC = 1 kHz and C 1 = 47 nF, R 1 calculates to:
R 1
a 1
2 $fcC 1
1 2 $·10^3 Hz·47·10$^9 F
3.38 k!
However, to design the first stage of a third-order unity-gain Bessel low-pass filter, assum- ing the same values for fC and C 1 , requires a different value for R 1. In this case, obtain a 1 for a third-order Bessel filter from Table 16–4 in Section 16.9 (Bessel coefficients) to calculate R 1 :
R 1
a 1
2 $fcC 1
2 $·10^3 Hz·47·10$^9 F
2.56 k!
When operating at unity gain, the noninverting amplifier reduces to a voltage follower (Fig- ure 16–14), thus inherently providing a superior gain accuracy. In the case of the inverting amplifier, the accuracy of the unity gain depends on the tolerance of the two resistors, R 1 and R 2.
R 1
C 1
VIN
VOUT
Figure 16–14. First-Order Noninverting Low-Pass Filter with Unity Gain
16.3.2 Second-Order Low-Pass Filter
There are two topologies for a second-order low-pass filter, the Sallen-Key and the Multi- ple Feedback (MFB) topology.
16.3.2.1 Sallen-Key Topology
The general Sallen-Key topology in Figure 16–15 allows for separate gain setting via A 0 = 1+R 4 /R 3. However, the unity-gain topology in Figure 16–16 is usually applied in filter designs with high gain accuracy, unity gain, and low Qs (Q < 3).
Low-Pass Filter Design
In order to obtain real values under the square root, C 2 must satisfy the following condi- tion:
C 2 ) C 1
4b 1
a 12
Example 16–2. Second-Order Unity-Gain Tschebyscheff Low-Pass Filter
The task is to design a second-order unity-gain Tschebyscheff low-pass filter with a corner frequency of fC = 3 kHz and a 3-dB passband ripple.
From Table 16–9 (the Tschebyscheff coefficients for 3-dB ripple), obtain the coefficients a 1 and b 1 for a second-order filter with a 1 = 1.0650 and b 1 = 1.9305.
Specifying C 1 as 22 nF yields in a C 2 of:
C 2 ) C 1
4b 1
a 12
22·10$^9 nF ·
4 ·1. 1.065^2
Inserting a 1 and b 1 into the resistor equation for R1,2 results in:
R 1
1.065·150·10$^9 $ !1.065·150·10$^9 "
$ 4·1.9305·22·10$^9 ·150·10$^9
4 $·3·10^3 ·22·10$^9 ·150·10$^9
1.26 k!
and
R 2
1.065·150·10$^9 !1.065·150·10$^9 "
$ 4·1.9305·22·10$^9 ·150·10$^9
4 $·3·10^3 ·22·10$^9 ·150·10$^9
1.30 k!
with the final circuit shown in Figure 16–17.
VIN
VOUT
1.26k 1.30k
22n
150n
Figure 16–17. Second-Order Unity-Gain Tschebyscheff Low-Pass with 3-dB Ripple
A special case of the general Sallen-Key topology is the application of equal resistor val- ues and equal capacitor values: R 1 = R 2 = R and C 1 = C 2 = C.
Low-Pass Filter Design
Active Filter Design Techniques 16-
The general transfer function changes to:
A(s)
A 0
1 (^) cRC! 3 $ A 0 "s ( (^) cRC)^2 s^2
A 0 1
R 4
R 3
with
The coefficient comparison with Equation 16–2 yields:
a 1 (^) cRC! 3 $ A 0 "
b 1 !^ cRC"
Given C and solving for R and A 0 results in:
R
b 1
2 $fcC
A 0 3 $
a 1
b 1
3 $
1 and Q
Thus, A 0 depends solely on the pole quality Q and vice versa; Q, and with it the filter type, is determined by the gain setting of A 0 :
Q
1 3 $ A 0
The circuit in Figure 16–18 allows the filter type to be changed through the various resistor ratios R 4 /R 3.
VIN
VOUT
R R
C
C
R 3
R 4
Figure 16–18. Adjustable Second-Order Low-Pass Filter
Table 16–1 lists the coefficients of a second-order filter for each filter type and gives the resistor ratios that adjust the Q.
Table 16–1. Second-Order FIlter Coefficients
SECOND-ORDER BESSEL BUTTERWORTH 3-dB TSCHEBYSCHEFF
a 1 1.3617 1.4142 1. b 1 0.618 1 1.
Q 0.58 0.71 1.
R 4 /R 3 0.268 0.568 0.
Low-Pass Filter Design
Active Filter Design Techniques 16-
In order to obtain real values for R 2 , C 2 must satisfy the following condition:
C 2 $ C 1
4b 1! 1 A 0 "
a 12
16.3.3 Higher-Order Low-Pass Filters
Higher-order low-pass filters are required to sharpen a desired filter characteristic. For that purpose, first-order and second-order filter stages are connected in series, so that the product of the individual frequency responses results in the optimized frequency re- sponse of the overall filter.
In order to simplify the design of the partial filters, the coefficients ai and bi for each filter type are listed in the coefficient tables (Tables 16–4 through 16–10 in Section 16.9), with each table providing sets of coefficients for the first 10 filter orders.
Example 16–3. Fifth-Order Filter
The task is to design a fifth-order unity-gain Butterworth low-pass filter with the corner fre- quency fC = 50 kHz.
First the coefficients for a fifth-order Butterworth filter are obtained from Table 16–5, Sec- tion 16.9:
ai bi
Filter 1 a 1 = 1 b 1 = 0
Filter 2 a 2 = 1.6180 b 2 = 1
Filter 3 a 3 = 0.6180 b 3 = 1
Then dimension each partial filter by specifying the capacitor values and calculating the required resistor values.
First Filter
R 1
C 1
VIN
VOUT
Figure 16–20. First-Order Unity-Gain Low-Pass
With C 1 = 1nF,
R 1
a 1
2 !fcC 1
1 2 !·50·10^3 Hz·1·10 9 F
3.18 k"
The closest 1% value is 3.16 kΩ.
Low-Pass Filter Design
Second Filter
VIN
VOUT
R 1 R 2
C 1
C 2
Figure 16–21. Second-Order Unity-Gain Sallen-Key Low-Pass Filter
With C 1 = 820 pF,
C 2 $ C 1
4b 2
a 22
820·10 12 F·
4· 1.618^2
1.26 nF
The closest 5% value is 1.5 nF.
With C 1 = 820 pF and C 2 = 1.5 nF, calculate the values for R1 and R2 through:
R 1
a 2 C 2 a 22 C 2
4b 2 C 1 C 2
4 !fcC 1 C 2
R 1
a 2 C 2 a 22 C 2
4b 2 C 1 C 2
and 4 !fcC 1 C 2
and obtain
R 1
1.618·1.5·10 9 !1.618·1.5·10 9 "
4·1·820·10 12 ·1.5·10 9
4 !·50·10^3 ·820·10 12 ·1.5·10 9
1.87 k"
R 2
1.618·1.5·10 9 !1.618·1.5·10 9 "
4·1·820·10 12 ·1.5·10 9
4 !·50·10^3 ·820·10 12 ·1.5·10 9
4.42 k"
R 1 and R 2 are available 1% resistors.
Third Filter
The calculation of the third filter is identical to the calculation of the second filter, except that a 2 and b 2 are replaced by a 3 and b 3 , thus resulting in different capacitor and resistor values.
Specify C 1 as 330 pF, and obtain C 2 with:
C 2 $ C 1
4b 3
a 32
330·10 12 F·
4· 0.618^2
3.46 nF
The closest 10% value is 4.7 nF.
