Chapter 16. Active Filter Design Techniques. Excerpted from Op Amps for Everyone. Literature Number SLOA088. Literature Number: SLOD006A


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1 hapter 16 Active Filter Design Techniques Literature Number SLOA088 Excerpted from Op Amps for Everyone Literature Number: SLOD006A
2 hapter 16 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, bandpass filters are used in the audio frequency range (0 khz to 20 khz) for modems and speech processing. Highfrequency bandpass filters (several hundred MHz) are used for channel selection in telephone central offices. Data acquisition systems usually require antialiasing lowpass filters as well as lowpass noise filters in their preceding signal conditioning stages. System power supplies often use bandrejection filters to suppress the 60Hz 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 allpass filters. At high frequencies (> 1 MHz), all of these filters usually consist of passive components such as inductors (L), resistors (R), and capacitors (). They are then called LR 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 operational amplifier (op amp) as the active device in combination with some resistors and capacitors to provide an LRlike filter performance at low frequencies (Figure 16 1). L R V OUT V OUT 1 Figure SecondOrder Passive LowPass and SecondOrder Active LowPass 161
3 Fundamentals of LowPass Filters This chapter covers active filters. It introduces the three main filter optimizations (Butterworth, Tschebyscheff, and Bessel), followed by five sections describing the most common active filter applications: lowpass, highpass, bandpass, bandrejection, and allpass filters. Rather than resembling just another filter book, the individual filter sections are written 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 calculate the individual circuit components. The chapter closes with a section on practical design hints for singlesupply filter designs Fundamentals of LowPass Filters The most simple lowpass filter is the passive R lowpass network shown in Figure R V OUT Figure FirstOrder Passive R LowPass Its transfer function is: A(s) 1 R s sr R where the complex frequency variable, s = jω+σ, allows for any time variable signals. For pure sine waves, the damping constant, σ, becomes zero and s = jω. For a normalized presentation of the transfer function, s is referred to the filter s corner frequency, or 3 db frequency, ω, and has these relationships: s s j j f j f With the corner frequency of the lowpass in Figure 16 2 being f = 1/2πR, s becomes s = sr and the transfer function A(s) results in: A(s) 1 1 s The magnitude of the gain response is: A 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 To avoid loading effects, op amps, operating as impedance converters, separate the individual filter stages. 162
4 Fundamentals of LowPass Filters R R R R V OUT Figure FourthOrder Passive R LowPass with Decoupling Amplifiers The resulting transfer function is: A(s) s 1 2 s (1 n s) In the case that all filters have the same cutoff frequency, f, the coefficients become 1 2 n n 2 1, and f of each partial filter is 1/α times higher than f of the overall filter. Figure 16 4 shows the results of a fourthorder R lowpass filter. The rolloff of each partial filter (urve 1) is 20 db/decade, increasing the rolloff of the overall filter (urve 2) to 80 db/decade. Note: Filter response graphs plot gain versus the normalized frequency axis Ω (Ω = f/f ). Active Filter Design Techniques 163
5 Fundamentals of LowPass Filters 0 10 A Gain db st Order Lowpass 4th Order Lowpass Ideal 4th Order Lowpass Frequency Ω φ Phase degrees Ideal 4th Order Lowpass 4th Order Lowpass 1st Order Lowpass Frequency Ω 100 Note: urve 1: 1storder partial lowpass filter, urve 2: 4thorder overall lowpass filter, urve 3: Ideal 4thorder lowpass filter Figure Frequency and Phase Responses of a FourthOrder Passive R LowPass 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. 164
6 Fundamentals of LowPass Filters A(s) In addition, Figure 16 4 shows the transfer function of an ideal fourthorder lowpass function (urve 3). In comparison to the ideal lowpass, the R lowpass lacks in the following characteristics: The passband gain varies long before the corner frequency, f, 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 80dB roll off by 1.5 octaves above f. The phase response is not linear, thus increasing the amount of signal distortion significantly. The gain and phase response of a lowpass filter can be optimized to satisfy one of the following three criteria: 1) A maximum passband flatness, 2) An immediate passbandtostopband transition, 3) A linear phase response. For that purpose, the transfer function must allow for complex poles and needs to be of the following type: A 0 1 a1 s b 1 s 2 1 a2 s b 2 s 2 1 an s b n s 2 A 0 i 1 ai s b i s 2 where A 0 is the passband gain at dc, and a i and b i are the filter coefficients. Since the denominator is a product of quadratic terms, the transfer function represents a series of cascaded secondorder lowpass stages, with a i and b i being positive real coefficients. These coefficients define the complex pole locations for each secondorder 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 f The transfer function of a passive R filter does not allow further optimization, due to the lack of complex poles. The only possibility to produce conjugate complex poles using pas Active Filter Design Techniques 165
7 Fundamentals of LowPass Filters sive components is the application of LR filters. However, these filters are mainly used at high frequencies. In the lower frequency range (< 10 MHz) the inductor values become very large and the filter becomes uneconomical to manufacture. In these cases active filters are used. Active filters are R networks that include an active device, such as an operational amplifier (op amp). Section 16.3 shows that the products of the R values and the corner frequency must yield the predetermined filter coefficients a i and b i, to generate the desired transfer function. The following paragraphs introduce the most commonly used filter optimizations Butterworth LowPass FIlters The Butterworth lowpass filter provides maximum passband flatness. Therefore, a Butterworth lowpass is often used as antialiasing filter in data converter applications where precise signal levels are required across the entire passband. Figure 16 5 plots the gain response of different orders of Butterworth lowpass filters versus the normalized frequency axis, Ω (Ω = f / f ); the higher the filter order, the longer the passband flatness A Gain db st Order 2nd Order 4th Order 10th Order Frequency Ω 100 Figure Amplitude Responses of Butterworth LowPass Filters 166
8 Fundamentals of LowPass Filters Tschebyscheff LowPass Filters The Tschebyscheff lowpass filters provide an even higher gain rolloff above f. 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 A Gain db th Order 2nd Order 4th Order Frequency Ω 100 Figure Gain Responses of Tschebyscheff LowPass Filters With increasing filter order, the influence of the ripple magnitude on the filter rolloff diminishes. Each ripple accounts for one secondorder filter stage. Filters with even order numbers generate ripples above the 0dB 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 Bessel LowPass Filters The Bessel lowpass filters have a linear phase response (Figure 16 7) over a wide frequency range, which results in a constant group delay (Figure 16 8) in that frequency range. Bessel lowpass filters, therefore, provide an optimum squarewave transmission behavior. However, the passband gain of a Bessel lowpass filter is not as flat as that of the Butterworth lowpass, and the transition from passband to stopband is by far not as sharp as that of a Tschebyscheff lowpass filter (Figure 16 9). Active Filter Design Techniques 167
9 Fundamentals of LowPass Filters 0 φ Phase degrees Bessel Butterworth Tschebyscheff Frequency Ω 100 Figure omparison of Phase Responses of FourthOrder LowPass Filters 1.4 Tgr Normalized Group Delay s/s Tschebyscheff Bessel Butterworth Frequency Ω 100 Figure omparison of Normalized Group Delay (Tgr) of FourthOrder LowPass Filters 168
10 Fundamentals of LowPass Filters A Gain db Butterworth Bessel Tschebyscheff Frequency Ω Figure omparison of Gain Responses of FourthOrder LowPass Filters Quality Factor Q The quality factor Q is an equivalent design parameter to the filter order n. Instead of designing an n th order Tschebyscheff lowpass, the problem can be expressed as designing a Tschebyscheff lowpass filter with a certain Q. For bandpass filters, Q is defined as the ratio of the mid frequency, f m, to the bandwidth at the two 3 db points: Q f m (f 2 f 1 ) For lowpass and highpass filters, Q represents the pole quality and is defined as: Q b i a i High Qs can be graphically presented as the distance between the 0dB line and the peak point of the filter s gain response. An example is given in Figure 16 10, which shows a tenthorder Tschebyscheff lowpass filter and its five partial filters with their individual Qs. Active Filter Design Techniques 169
11 Fundamentals of LowPass Filters A Gain db 20 Overall Filter Q st Stage 10 2nd Stage 3rd Stage 20 4th Stage 5th Stage Frequency Ω Figure Graphical Presentation of Quality Factor Q on a TenthOrder Tschebyscheff LowPass Filter with 3dB 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 which is within 1% of the theoretical value of Q = 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 Summary The general transfer function of a lowpass filter is : A(s) A 0 i 1 ai s b i s 2 (16 1) The filter coefficients a i and b i distinguish between Butterworth, Tschebyscheff, and Bessel filters. The coefficients for all three types of filters are tabulated down to the tenth order in Section 16.9, Tables 16 4 through
12 LowPass Filter Design 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 f with n 20 dbdecade, a i and b i determine the gain behavior in the passband. In addition, the ratio bi a i more a filter inclines to instability. Q is defined as the pole quality. The higher the Q value, the 16.3 LowPass Filter Design Equation 16 1 represents a cascade of secondorder lowpass filters. The transfer function of a single stage is: A i (s) A 0 1 ai s b i s 2 (16 2) For a firstorder filter, the coefficient b is always zero (b 1 =0), thus yielding: A(s) A 0 1 a 1 s (16 3) The firstorder and secondorder filter stages are the building blocks for higherorder filters. Often the filters operate at unity gain (A 0 =1) to lessen the stringent demands on the op amp s openloop gain. Figure shows the cascading of filter stages up to the sixth order. A filter with an even order number consists of secondorder stages only, while filters with an odd order number include an additional firstorder stage at the beginning. Active Filter Design Techniques 1611
13 LowPass Filter Design 1st order 1st order a=1 2nd order 2nd order a 1, b 1 3rd order 1st order a 1 2nd order a 2, b 2 4th order 2nd order a 1, b 1 2nd order a 2, b 2 5th order 1st order a 1 2nd order a 2, b 2 2nd order a 3, b 3 6th order 2nd order a 1, b 1 2nd order a 2, b 2 2nd order a 3, b 3 Figure ascading Filter Stages for HigherOrder Filters Figure demonstrated that the higher the corner frequency of a partial filter, the higher its Q. Therefore, to avoid the saturation of the individual stages, the filters need to be placed in the order of rising Q values. The Q values for each filter order are listed (in rising order) in Section 16 9, Tables 16 4 through FirstOrder LowPass Filter Figures and show a firstorder lowpass filter in the inverting and in the noninverting configuration. V OUT 1 R 3 Figure FirstOrder Noninverting LowPass Filter 1612
14 LowPass Filter Design 1 V OUT Figure FirstOrder Inverting LowPass Filter The transfer functions of the circuits are: A(s) 1 R 3 1 c 1 s and A(s) 1 c 1 s 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 and A R 0 3 and a 1 c 1 a 1 c 1 To dimension the circuit, specify the corner frequency (f ), the dc gain (A 0 ), and capacitor 1, and then solve for resistors and : a 1 and R 2f c a 1 1 2f c 1 R 3 and A0 1 A 0 The coefficient a 1 is taken from one of the coefficient tables, Tables 16 4 through in Section 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 firstorder stage is different from the corner frequency of the overall filter. Active Filter Design Techniques 1613
15 LowPass Filter Design Example FirstOrder UnityGain LowPass Filter For a firstorder unitygain lowpass filter with f = 1 khz and 1 = 47 nf, calculates to: a 1 1 2f c Hz k F However, to design the first stage of a thirdorder unitygain Bessel lowpass filter, assuming the same values for f and 1, requires a different value for. In this case, obtain a 1 for a thirdorder Bessel filter from Table 16 4 in Section 16.9 (Bessel coefficients) to calculate : a f c Hz k F When operating at unity gain, the noninverting amplifier reduces to a voltage follower (Figure 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, and. V OUT 1 Figure FirstOrder Noninverting LowPass Filter with Unity Gain SecondOrder LowPass Filter There are two topologies for a secondorder lowpass filter, the SallenKey and the Multiple Feedback (MFB) topology SallenKey Topology The general SallenKey topology in Figure allows for separate gain setting via A 0 = 1+R 4 /R 3. However, the unitygain topology in Figure is usually applied in filter designs with high gain accuracy, unity gain, and low Qs (Q < 3)
16 LowPass Filter Design V OUT 1 R 3 R 4 Figure General SallenKey LowPass Filter V OUT 1 Figure UnityGain SallenKey LowPass Filter The transfer function of the circuit in Figure is: A(s) A 0 1 c 1 R1 1 A0 R1 s c 2 1 s 2 For the unitygain circuit in Figure (A 0 =1), the transfer function simplifies to: A(s) 1 1 c 1 R1 s c 2 1 s 2 The coefficient comparison between this transfer function and Equation 16 2 yields: A 0 1 a 1 c 1 R1 b 1 c 2 1 Given 1 and, the resistor values for and are calculated through:,2 a 1 a b1 1 4f c 1 Active Filter Design Techniques 1615
17 LowPass Filter Design In order to obtain real values under the square root, must satisfy the following condition: 1 4b 1 a 1 2 Example SecondOrder UnityGain Tschebyscheff LowPass Filter The task is to design a secondorder unitygain Tschebyscheff lowpass filter with a corner frequency of f = 3 khz and a 3dB passband ripple. From Table 16 9 (the Tschebyscheff coefficients for 3dB ripple), obtain the coefficients a 1 and b 1 for a secondorder filter with a 1 = and b 1 = Specifying 1 as 22 nf yields in a of: 1 4b 1 a nf nf Inserting a 1 and b 1 into the resistor equation for,2 results in: k and k with the final circuit shown in Figure n 1.26k 1.30k 22n V OUT Figure SecondOrder UnityGain Tschebyscheff LowPass with 3dB Ripple A special case of the general SallenKey topology is the application of equal resistor values and equal capacitor values: = = R and 1 = =
18 LowPass Filter Design The general transfer function changes to: A(s) A 0 1 c R 3 A0 s (c R) 2 s 2 A 0 1 R 4 with R 3 The coefficient comparison with Equation 16 2 yields: a 1 c R 3 A0 b 1 c R Given and solving for R and A 0 results in: R b 1 2f c and A 0 3 a 1 b 3 1 Q 1 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 allows the filter type to be changed through the various resistor ratios R 4 /R 3. R R V OUT R 3 R 4 Figure Adjustable SecondOrder LowPass Filter Table 16 1 lists the coefficients of a secondorder filter for each filter type and gives the resistor ratios that adjust the Q. Table SecondOrder FIlter oefficients SEONDORDER BESSEL BUTTERWORTH 3dB TSHEBYSHEFF a b Q R4/R Active Filter Design Techniques 1617
19 LowPass Filter Design Multiple Feedback Topology The MFB topology is commonly used in filters that have high Qs and require a high gain. R 3 1 V OUT Figure SecondOrder MFB LowPass Filter The transfer function of the circuit in Figure is: A(s) 1 c 1 R 3 R 3 s c 2 1 R 3 s 2 Through coefficient comparison with Equation 16 2 one obtains the relation: A 0 a 1 c 1 R 3 R 3 b 1 c 2 1 R 3 Given 1 and, and solving for the resistors R 3 : a 1 A 0 b 1 R f c 1 a b1 1 1 A0 4f c
20 LowPass Filter Design In order to obtain real values for, must satisfy the following condition: 1 4b 1 1 A0 a HigherOrder LowPass Filters Higherorder lowpass filters are required to sharpen a desired filter characteristic. For that purpose, firstorder and secondorder filter stages are connected in series, so that the product of the individual frequency responses results in the optimized frequency response of the overall filter. In order to simplify the design of the partial filters, the coefficients a i and b i for each filter type are listed in the coefficient tables (Tables 16 4 through in Section 16.9), with each table providing sets of coefficients for the first 10 filter orders. Example FifthOrder Filter The task is to design a fifthorder unitygain Butterworth lowpass filter with the corner frequency f = 50 khz. First the coefficients for a fifthorder Butterworth filter are obtained from Table 16 5, Section 16.9: a i b i Filter 1 a 1 = 1 b 1 = 0 Filter 2 a 2 = b 2 = 1 Filter 3 a 3 = b 3 = 1 Then dimension each partial filter by specifying the capacitor values and calculating the required resistor values. First Filter V OUT 1 Figure FirstOrder UnityGain LowPass With 1 = 1nF, a 1 1 2f c Hz k F The closest 1% value is 3.16 kω. Active Filter Design Techniques 1619
21 LowPass Filter Design Second Filter V OUT 1 Figure SecondOrder UnityGain SallenKey LowPass Filter With 1 = 820 pf, 4b 1 2 a F nf The closest 5% value is 1.5 nf. With 1 = 820 pf and = 1.5 nf, calculate the values for R1 and R2 through: a 2 a b2 1 R 4fc a 2 a b2 1 and 4f c 1 Third Filter and obtain and are available 1% resistors k 4.42 k 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 1 as 330 pf, and obtain with: 4b a F nf The closest 10% value is 4.7 nf
22 HighPass Filter Design With 1 = 330 pf and = 4.7 nf, the values for R1 and R2 are: = 1.45 kω, with the closest 1% value being 1.47 kω = 4.51 kω, with the closest 1% value being 4.53 kω Figure shows the final filter circuit with its partial filter stages. 3.16k 1.87k 4.42k 1n 820p 1.5n 1.47k 4.53k 330p 4.7n V OUT Figure FifthOrder UnityGain Butterworth LowPass Filter 16.4 HighPass Filter Design By replacing the resistors of a lowpass filter with capacitors, and its capacitors with resistors, a highpass filter is created. 1 V OUT 1 V OUT Figure LowPass to HighPass Transition Through omponents Exchange To plot the gain response of a highpass filter, mirror the gain response of a lowpass filter at the corner frequency, Ω=1, thus replacing Ω with 1/Ω and S with 1/S in Equation Active Filter Design Techniques 1621
23 HighPass Filter Design 10 A0 A A Gain db 0 10 Lowpass Highpass Frequency Ω Figure Developing The Gain Response of a HighPass Filter The general transfer function of a highpass filter is then: A(s) A i 1 a i s b i s 2 (16 4) with A being the passband gain. Since Equation 16 4 represents a cascade of secondorder highpass filters, the transfer function of a single stage is: A i (s) A 1 a i s b i s 2 (16 5) With b=0 for all firstorder filters, the transfer function of a firstorder filter simplifies to: A(s) A 0 1 a i s (16 6) 1622
24 HighPass Filter Design FirstOrder HighPass Filter Figure and show a firstorder highpass filter in the noninverting and the inverting configuration. 1 V OUT R 3 Figure FirstOrder Noninverting HighPass Filter 1 V OUT Figure FirstOrder Inverting HighPass Filter The transfer functions of the circuits are: A(s) 1 R c 1 1s and A(s) 1 1 c 1 1s 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 6 provides two different passband gain factors: A 1 and A R 3 while the term for the coefficient a 1 is the same for both circuits: a 1 1 c 1 Active Filter Design Techniques 1623
25 HighPass Filter Design To dimension the circuit, specify the corner frequency (f ), the dc gain (A ), and capacitor ( 1 ), and then solve for and : 1 2f c a 1 1 R 3 (A 1) and A SecondOrder HighPass Filter Highpass filters use the same two topologies as the lowpass filters: SallenKey and Multiple Feedback. The only difference is that the positions of the resistors and the capacitors have changed SallenKey Topology The general SallenKey topology in Figure allows for separate gain setting via A 0 = 1+R 4 /R 3. 1 V OUT R 3 R 4 Figure General SallenKey HighPass Filter A(s) The transfer function of the circuit in Figure is: 1 R 2 1 R1 (1) 1s c R 4 with 2 2 c 1 s 3 The unitygain topology in Figure is usually applied in lowq filters with high gain accuracy. V OUT Figure UnityGain SallenKey HighPass Filter 1624
26 HighPass Filter Design To simplify the circuit design, it is common to choose unitygain (α = 1), and 1 = =. The transfer function of the circuit in Figure then simplifies to: A(s) c 1s c s 2 The coefficient comparison between this transfer function and Equation 16 5 yields: A 1 a 1 2 c b c Given, the resistor values for and are calculated through: 1 f c a 1 a 1 4f c b Multiple Feedback Topology The MFB topology is commonly used in filters that have high Qs and require a high gain. To simplify the computation of the circuit, capacitors 1 and 3 assume the same value ( 1 = 3 = ) as shown in Figure = 3 = V OUT Figure SecondOrder MFB HighPass Filter The transfer function of the circuit in Figure is: A(s) 1 2 c 1s 2 c 1 s 2 Active Filter Design Techniques 1625
27 HighPass Filter Design Through coefficient comparison with Equation 16 5, obtain the following relations: A a 1 2 c b 1 2 c Given capacitors and, and solving for resistors and : 1 2A 2f c a 1 a 1 2f c b 1 (1 2A ) The passband gain (A ) of a MFB highpass filter can vary significantly due to the wide tolerances of the two capacitors and. To keep the gain variation at a minimum, it is necessary to use capacitors with tight tolerance values HigherOrder HighPass Filter Likewise, as with the lowpass filters, higherorder highpass filters are designed by cascading firstorder and secondorder filter stages. The filter coefficients are the same ones used for the lowpass filter design, and are listed in the coefficient tables (Tables 16 4 through in Section 16.9). Example ThirdOrder HighPass Filter with f = 1 khz The task is to design a thirdorder unitygain Bessel highpass filter with the corner frequency f = 1 khz. Obtain the coefficients for a thirdorder Bessel filter from Table 16 4, Section 16.9: a i b i Filter 1 a 1 = b 1 = 0 Filter 2 a 2 = b 2 = and compute each partial filter by specifying the capacitor values and calculating the required resistor values. First Filter With 1 = 100 nf, 1626
28 BandPass Filter Design 1 1 2f c a Hz k F losest 1% value is 2.1 kω. Second Filter With = 100nF, 1 f c a 1 losest 1% value is 3.16 kω. a 1 4f c b 1 losest 1% value is 1.65 kω. Figure shows the final filter circuit k k n 100n 100n 1.65k 2.10k 3.16k VOUT Figure ThirdOrder UnityGain Bessel HighPass 16.5 BandPass Filter Design In Section 16.4, a highpass response was generated by replacing the term S in the lowpass transfer function with the transformation 1/S. Likewise, a bandpass characteristic is generated by replacing the S term with the transformation: 1 s 1 s (16 7) In this case, the passband characteristic of a lowpass filter is transformed into the upper passband half of a bandpass filter. The upper passband is then mirrored at the mid frequency, f m (Ω=1), into the lower passband half. Active Filter Design Techniques 1627
29 BandPass Filter Design A [db] 0 3 A [db] 0 3 Ω 0 1 Ω 0 Ω 1 1 Ω 2 Ω Figure LowPass to BandPass Transition The corner frequency of the lowpass filter transforms to the lower and upper 3 db frequencies of the bandpass, Ω 1 and Ω 2. The difference between both frequencies is defined as the normalized bandwidth Ω: 2 1 The normalized mid frequency, where Q = 1, is: m In analogy to the resonant circuits, the quality factor Q is defined as the ratio of the mid frequency (f m ) to the bandwidth (B): Q f m B f m 1 1 f 2 f (16 8) The simplest design of a bandpass filter is the connection of a highpass filter and a lowpass filter in series, which is commonly done in wideband filter applications. Thus, a firstorder highpass and a firstorder lowpass provide a secondorder bandpass, while a secondorder highpass and a secondorder lowpass result in a fourthorder bandpass response. In comparison to wideband filters, narrowband filters of higher order consist of cascaded secondorder bandpass filters that use the SallenKey or the Multiple Feedback (MFB) topology
30 BandPass Filter Design SecondOrder BandPass Filter To develop the frequency response of a secondorder bandpass filter, apply the transformation in Equation 16 7 to a firstorder lowpass transfer function: Replacing s with A(s) A 0 1 s 1 s 1 s yields the general transfer function for a secondorder bandpass filter: A(s) A0 s 1 s s 2 (16 9) When designing bandpass filters, the parameters of interest are the gain at the mid frequency (A m ) and the quality factor (Q), which represents the selectivity of a bandpass filter. Therefore, replace A 0 with A m and Ω with 1/Q (Equation 16 7) and obtain: A(s) A m Q s 1 1 s s2 Q (16 10) Figure shows the normalized gain response of a secondorder bandpass filter for different Qs Q = 1 A Gain db Q = Frequency Ω Figure Gain Response of a SecondOrder BandPass Filter Active Filter Design Techniques 1629
31 BandPass Filter Design The graph shows that the frequency response of secondorder bandpass filters gets steeper with rising Q, thus making the filter more selective SallenKey Topology R R 2R V OUT Figure SallenKey BandPass The SallenKey bandpass circuit in Figure has the following transfer function: A(s) G R m s 1 R m (3 G) s m2 s 2 Through coefficient comparison with Equation 16 10, obtain the following equations: midfrequency: inner gain: gain at f m: filter quality: f m 1 2R G 1 A m G 3 G Q 1 3 G The SallenKey circuit has the advantage that the quality factor (Q) can be varied via the inner gain (G) without modifying the mid frequency (fm). A drawback is, however, that Q and A m cannot be adjusted independently. are must be taken when G approaches the value of 3, because then A m becomes infinite and causes the circuit to oscillate. To set the mid frequency of the bandpass, specify f m and and then solve for R: R 1 2f m 1630
32 BandPass Filter Design Because of the dependency between Q and A m, there are two options to solve for : either to set the gain at mid frequency: 2A m 1 1 A m or to design for a specified Q: 2Q 1 Q Multiple Feedback Topology V OUT R 3 Figure MFB BandPass The MFB bandpass circuit in Figure has the following transfer function: A(s) R 3 R 3 m s 1 2 R 3 R 3 m s R 3 R 3 m2 s 2 The coefficient comparison with Equation 16 9, yields the following equations: midfrequency: f m 1 R 3 2 R1 R 3 gain at f m : filter quality: bandwidth: A m 2 Q f m B 1 The MFB bandpass allows to adjust Q, A m, and f m independently. Bandwidth and gain factor do not depend on R 3. Therefore, R 3 can be used to modify the mid frequency with Active Filter Design Techniques 1631
33 BandPass Filter Design out affecting bandwidth, B, or gain, A m. For low values of Q, the filter can work without R3, however, Q then depends on A m via: A m 2Q 2 Example SecondOrder MFB BandPass Filter with f m = 1 khz To design a secondorder MFB bandpass filter with a mid frequency of f m = 1 khz, a quality factor of Q = 10, and a gain of A m = 2, assume a capacitor value of = 100 nf, and solve the previous equations for through R 3 in the following sequence: Q f m k 1 khz 100 nf 31.8 k 7.96 k 2A m 4 R 3 A m 2Q k A m FourthOrder BandPass Filter (Staggered Tuning) Figure shows that the frequency response of secondorder bandpass filters gets steeper with rising Q. However, there are bandpass applications that require a flat gain response close to the mid frequency as well as a sharp passbandtostopband transition. These tasks can be accomplished by higherorder bandpass filters. Of particular interest is the application of the lowpass to bandpass transformation onto a secondorder lowpass filter, since it leads to a fourthorder bandpass filter. Replacing the S term in Equation 16 2 with Equation 16 7 gives the general transfer function of a fourthorder bandpass: A(s) s 2 A 0 () 2 b 1 1 a 1 b 1 s 2 ()2 b 1 s 2 a 1 b 1 s 3 s 4 (16 11) Similar to the lowpass filters, the fourthorder transfer function is split into two secondorder bandpass terms. Further mathematical modifications yield: A mi A s mi s Q A(s) i 1 s (s) 2 Q 1 Q i 1 1 s Q s2 i (16 12) Equation represents the connection of two secondorder bandpass filters in series, where 1632
34 BandPass Filter Design A mi is the gain at the mid frequency, f mi, of each partial filter Q i is the pole quality of each filter α and 1/α are the factors by which the mid frequencies of the individual filters, f m1 and f m2, derive from the mid frequency, f m, of the overall bandpass. In a fourthorder bandpass filter with high Q, the mid frequencies of the two partial filters differ only slightly from the overall mid frequency. This method is called staggered tuning. Factor α needs to be determined through successive approximation, using equation 16 13: 2 a 1 b () 2 b 1 0 (16 13) with a 1 and b 1 being the secondorder lowpass coefficients of the desired filter type. To simplify the filter design, Table 16 2 lists those coefficients, and provides the α values for three different quality factors, Q = 1, Q = 10, and Q = 100. Table Values of α For Different Filter Types and Different Qs Bessel Butterworth Tschebyscheff a a a b b b Q Q Q Ω Ω Ω α α α Active Filter Design Techniques 1633
35 BandPass Filter Design After α has been determined, all quantities of the partial filters can be calculated using the following equations: The mid frequency of filter 1 is: f m1 f m (16 14) the mid frequency of filter 2 is: f m2 f m (16 15) with f m being the mid frequency of the overall forthorder bandpass filter. The individual pole quality, Q i, is the same for both filters: Q i Q 1 2 b 1 a 1 (16 16) with Q being the quality factor of the overall filter. The individual gain (A mi ) at the partial mid frequencies, f m1 and f m2, is the same for both filters: A mi Q i Q A m B1 (16 17) with A m being the gain at mid frequency, f m, of the overall filter. Example FourthOrder Butterworth BandPass Filter The task is to design a fourthorder Butterworth bandpass with the following parameters: mid frequency, f m = 10 khz bandwidth, B = 1000 Hz and gain, A m = 1 From Table 16 2 the following values are obtained: a1 = b1 = 1 α =
36 BandPass Filter Design In accordance with Equations and 16 15, the mid frequencies for the partial filters are: f mi 10 khz khz and f m2 10 khz khz The overall Q is defined as Q f m B, and for this example results in Q = 10. Using Equation 16 16, the Q i of both filters is: Q i With Equation 16 17, the passband gain of the partial filters at f m1 and f m2 calculates to: A mi The Equations and show that Q i and A mi of the partial filters need to be independently adjusted. The only circuit that accomplishes this task is the MFB bandpass filter in Paragraph To design the individual secondorder bandpass filters, specify = 10 nf, and insert the previously determined quantities for the partial filters into the resistor equations of the MFB bandpass filter. The resistor values for both partial filters are calculated below. Filter 1: Filter 2: 1 Q i f m khz 10 nf 46.7 k 2 Q i f m k khz 10 nf 1 1 2A mi R 31 A mi1 2Q i 2 A mi 46.7 k k 2 2 2A mi k R 32 A mi2 2Q 2 i A mi 43.5 k 15.4 k k Figure compares the gain response of a fourthorder Butterworth bandpass filter with Q = 1 and its partial filters to the fourthorder gain of Example 16 4 with Q = 10. Active Filter Design Techniques 1635
37 BandRejection Filter Design A1 A2 Q = 1 Q = 10 A Gain db k 10 k 100 k f Frequency Hz 1 M Figure Gain Responses of a FourthOrder Butterworth BandPass and its Partial Filters 16.6 BandRejection Filter Design A bandrejection filter is used to suppress a certain frequency rather than a range of frequencies. Two of the most popular bandrejection filters are the active twint and the active Wien Robinson circuit, both of which are secondorder filters. To generate the transfer function of a secondorder bandrejection filter, replace the S term of a firstorder lowpass response with the transformation in 16 18: s 1 s (16 18) which gives: A(s) A 0 1 s 2 1 s s 2 (16 19) Thus the passband characteristic of the lowpass filter is transformed into the lower passband of the bandrejection filter. The lower passband is then mirrored at the mid frequency, f m (Ω=1), into the upper passband half (Figure 16 36)
38 BandRejection Filter Design A [db] 0 3 A [db] 0 3 Ω 0 1 Ω 0 Ω 1 1 Ω 2 Ω Figure LowPass to BandRejection Transition The corner frequency of the lowpass transforms to the lower and upper 3dB frequencies of the bandrejection filter Ω 1 and Ω 2. The difference between both frequencies is the normalized bandwidth Ω: max min Identical to the selectivity of a bandpass filter, the quality of the filter rejection is defined as: Q f m B 1 Therefore, replacing Ω in Equation with 1/Q yields: A(s) A 0 1 s s s2 Q (16 20) Active TwinT Filter The original twint filter, shown in Figure 16 37, is a passive Rnetwork with a quality factor of Q = To increase Q, the passive filter is implemented into the feedback loop of an amplifier, thus turning into an active bandrejection filter, shown in Figure R/2 V OUT R R 2 Figure Passive TwinT Filter Active Filter Design Techniques 1637
39 BandRejection Filter Design R/2 R R 2 V OUT Figure Active TwinT Filter The transfer function of the active twint filter is: k1 s 2 A(s) 1 2(2 k) s s 2 (16 21) omparing the variables of Equation with Equation provides the equations that determine the filter parameters: midfrequency: inner gain: passband gain: f m 1 2R G 1 A 0 G rejection quality: Q 1 2(2 G) The twint circuit has the advantage that the quality factor (Q) can be varied via the inner gain (G) without modifying the mid frequency (f m ). However, Q and A m cannot be adjusted independently. To set the mid frequency of the bandpass, specify f m and, and then solve for R: R 1 2f m Because of the dependency between Q and A m, there are two options to solve for : either to set the gain at mid frequency: A0 1 R
40 BandRejection Filter Design or to design for a specific Q: 1 1 2Q Active WienRobinson Filter The WienRobinson bridge in Figure is a passive bandrejection filter with differential output. The output voltage is the difference between the potential of a constant voltage divider and the output of a bandpass filter. Its Qfactor is close to that of the twint circuit. To achieve higher values of Q, the filter is connected into the feedback loop of an amplifier. R 2 V OUT R Figure Passive WienRobinson Bridge R 3 2 R 4 R V OUT R Figure Active WienRobinson Filter The active WienRobinson filter in Figure has the transfer function: A(s) 1 s s s2 1 (16 22) with and R 3 R 4 omparing the variables of Equation with Equation provides the equations that determine the filter parameters: Active Filter Design Techniques 1639
41 BandRejection Filter Design midfrequency: passband gain: f m 1 2R A 0 1 rejection quality: Q 1 3 To calculate the individual component values, establish the following design procedure: 1) Define f m and and calculate R with: R 1 2f m 2) Specify Q and determine α via: 3Q 1 3) Specify A 0 and determine β via: A 0 3Q 4) Define and calculate R 3 and R 4 with: R 3 and R 4 In comparison to the twint circuit, the WienRobinson filter allows modification of the passband gain, A 0, without affecting the quality factor, Q. If f m is not completely suppressed due to component tolerances of R and, a finetuning of the resistor 2 is required. Figure shows a comparison between the filter response of a passive bandrejection filter with Q = 0.25, and an active secondorder filter with Q = 1, and Q =
42 AllPass Filter Design 0 A Gain db 5 10 Q = 10 Q = 1 Q = k 10 k Frequency Ω Figure omparison of Q Between Passive and Active BandRejection Filters 16.7 AllPass Filter Design In comparison to the previously discussed filters, an allpass filter has a constant gain across the entire frequency range, and a phase response that changes linearly with frequency. Because of these properties, allpass filters are used in phase compensation and signal delay circuits. Similar to the lowpass filters, allpass circuits of higher order consist of cascaded firstorder and secondorder allpass stages. To develop the allpass transfer function from a lowpass response, replace A 0 with the conjugate complex denominator. The general transfer function of an allpass is then: i 1 ai s b i s 2 A(s) i 1 ai s b i s 2 (16 23) with a i and b i being the coefficients of a partial filter. The allpass coefficients are listed in Table of Section Expressing Equation in magnitude and phase yields: Active Filter Design Techniques 1641
43 AllPass Filter Design i 1 bi 2 2 a i 2 2 A(s) i 1 bi 2 2 a 2 i 2 e ja e ja (16 24) This gives a constant gain of 1, and a phase shift,φ, of: a 2 2 arctan i 1 b i 2 i (16 25) To transmit a signal with minimum phase distortion, the allpass filter must have a constant group delay across the specified frequency band. The group delay is the time by which the allpass filter delays each frequency within that band. The frequency at which the group delay drops to 1 2 times its initial value is the corner frequency, f. The group delay is defined through: t gr d d (16 26) To present the group delay in normalized form, refer t gr to the period of the corner frequency, T, of the allpass circuit: T gr t gr T c t gr f c t gr c 2 (16 27) Substituting t gr through Equation gives: T gr 1 2 d d (16 28) 1642
44 AllPass Filter Design Inserting the ϕ term in Equation into Equation and completing the derivation, results in: T gr 1 i a i 1 bi 2 1 a1 2 2b 1 2 b (16 29) Setting Ω = 0 in Equation gives the group delay for the low frequencies, 0 < Ω < 1, which is: T gr0 1 i a i (16 30) The values for T gr0 are listed in Table 16 10, Section 16.9, from the first to the tenth order. In addition, Figure shows the group delay response versus the frequency for the first ten orders of allpass filters. Tgr Normalized Group Delay s/s th Order 9th Order 8th Order 7th Order 6th Order 5th Order 4th Order 3rd Order 2nd Order 1st Order Frequency Ω 100 Figure Frequency Response of the Group Delay for the First 10 Filter Orders Active Filter Design Techniques 1643
45 AllPass Filter Design FirstOrder AllPass Filter Figure shows a firstorder allpass filter with a gain of +1 at low frequencies and a gain of 1 at high frequencies. Therefore, the magnitude of the gain is 1, while the phase changes from 0 to 180. V OUT R Figure FirstOrder AllPass The transfer function of the circuit above is: A(s) 1 R c s 1 R c s The coefficient comparison with Equation (b 1 =1), results in: a i R 2f c (16 31) To design a firstorder allpass, specify f and and then solve for R: R a i 2f c (16 32) Inserting Equation into and substituting ω with Equation provides the maximum group delay of a firstorder allpass filter: t gr0 2R (16 33) SecondOrder AllPass Filter Figure shows that one possible design for a secondorder allpass filter is to subtract the output voltage of a secondorder bandpass filter from its input voltage. R 3 R V OUT R Figure SecondOrder AllPass Filter 1644
46 AllPass Filter Design The transfer function of the circuit in Figure is: A(s) 1 2R1 c s c2 s c s c2 s 2 The coefficient comparison with Equation yields: a 1 4f c b 1 a 1 f c a 1 2 b 1 R R 3 (16 34) (16 35) (16 36) To design the circuit, specify f,, and R, and then solve for the resistor values: a 1 4f c b 1 a 1 f c R 3 R (16 37) (16 38) (16 39) Inserting Equation into Equation16 30 and substituting ω with Equation gives the maximum group delay of a secondorder allpass filter: t (16 40) gr HigherOrder AllPass Filter Higherorder allpass filters consist of cascaded firstorder and secondorder filter stages. Example ms Delay AllPass Filter A signal with the frequency spectrum, 0 < f < 1 khz, needs to be delayed by 2 ms. To keep the phase distortions at a minimum, the corner frequency of the allpass filter must be f 1 khz. Equation determines the normalized group delay for frequencies below 1 khz: T gro t gr0 T 2ms 1 khz 2.0 Figure confirms that a seventhorder allpass is needed to accomplish the desired delay. The exact value, however, is T gr0 = To set the group delay to precisely 2 ms, solve Equation for f and obtain the corner frequency: Active Filter Design Techniques 1645
47 AllPass Filter Design f T gr0 t gr khz To complete the design, look up the filter coefficients for a seventhorder allpass filter, specify, and calculate the resistor values for each partial filter. ascading the firstorder allpass with the three secondorder stages results in the desired seventhorder allpass filter R R 33 R 3 R R 34 R 4 V OUT R 4 Figure SeventhOrder AllPass Filter 1646
48 Practical Design Hints 16.8 Practical Design Hints This section introduces dcbiasing techniques for filter designs in singlesupply applications, which are usually not required when operating with dual supplies. It also provides recommendations on selecting the type and value range of capacitors and resistors as well as the decision criteria for choosing the correct op amp Filter ircuit Biasing The filter diagrams in this chapter are drawn for dual supply applications. The op amp operates from a positive and a negative supply, while the input and the output voltage are referenced to ground (Figure 16 46). +V 1 VOUT V Figure DualSupply Filter ircuit For the single supply circuit in Figure 16 47, the lowest supply voltage is ground. For a symmetrical output signal, the potential of the noninverting input is levelshifted to midrail. +V R B IN R B VMID VOUT Figure SingleSupply Filter ircuit The coupling capacitor, IN in Figure 16 47, accouples the filter, blocking any unknown dc level in the signal source. The voltage divider, consisting of the two equalbias resistors R B, divides the supply voltage to V MID and applies it to the inverting op amp input. For simple filter input structures, passive R networks often provide a lowcost biasing solution. In the case of more complex input structures, such as the input of a secondorder Active Filter Design Techniques 1647
49 +V +V Practical Design Hints lowpass filter, the R network can affect the filter characteristic. Then it is necessary to either include the biasing network into the filter calculations, or to insert an input buffer between biasing network and the actual filter circuit, as shown in Figure R B IN VMID VMID V MID VMID R B 1 V OUT Figure Biasing a SallenKey LowPass IN accouples the filter, blocking any dc level in the signal source. V MID is derived from V via the voltage divider. The op amp operates as a voltage follower and as an impedance converter. V MID is applied via the dc path, and, to the noninverting input of the filter amplifier. Note that the parallel circuit of the resistors, R B, together with IN create a highpass filter. To avoid any effect on the lowpass characteristic, the corner frequency of the input highpass must be low versus the corner frequency of the actual lowpass. The use of an input buffer causes no loading effects on the lowpass filter, thus keeping the filter calculation simple. In the case of a higherorder filter, all following filter stages receive their bias level from the preceding filter amplifier. Figure shows the biasing of an multiple feedback (MFB) lowpass filter
50 Practical Design Hints +V +V R B IN VMID VMID 1 R 3 R B V OUT R B +V V MID B R B V MID to further filter stages Figure Biasing a SecondOrder MFB LowPass Filter The input buffer decouples the filter from the signal source. The filter itself is biased via the noninverting amplifier input. For that purpose, the bias voltage is taken from the output of a V MID generator with low output impedance. The op amp operates as a difference amplifier and subtracts the bias voltage of the input buffer from the bias voltage of the V MID generator, thus yielding a dc potential of V MID at zero input signal. A lowcost alternative is to remove the op amp and to use a passive biasing network instead. However, to keep loading effects at a minimum, the values for R B must be significantly higher than without the op amp. The biasing of a SallenKey and an MFB highpass filter is shown in Figure The input capacitors of highpass filters already provide the accoupling between filter and signal source. Both circuits use the V MID generator from Figure for biasing. While the MFB circuit is biased at the noninverting amplifier input, the SallenKey highpass is biased via the only dc path available, which is. In the ac circuit, the input signals travel via the low output impedance of the op amp to ground. Active Filter Design Techniques 1649
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