Appendix B Higher Order Filters 21, List of Figures. 1 Low Pass Sallen Key Architecture 6, 2 Low Pass MFB Architecture 7. 3 Building Even Order Filters by Cascading Second Order Stages 8. 4 Building Odd Order Filters by Cascading Second Order Stages and Adding a Single Real Pole 8. 5 Sallen Key Circuit and Component Values fc 1 kHz 11. 6 MFB Circuit and Component Values fc 1 kHz 11, 7 Second Order Butterworth Filter Frequency Response 12. 8 Second Order Bessel Filter Frequency Response 12. 9 Second Order 3 dB Chebyshev Filter Frequency Response 13. 10 Second Order Butterworth Bessel and 3 dB Chebyshev Filter Frequency Response 13. 11 Transient Response of the Three Filters 14, 12 Second Order Low Pass Sallen Key High Frequency Model 14. 13 Sallen Key Butterworth Filter With RC Added in Series With the Output 15. 14 Second Order Low Pass MFB High Frequency Model 16. 15 MFB Butterworth Filter With RC Added in Series With the Output 16. B 1 Fifth Order Low Pass Filter Topology Cascading Two Sallen Key Stages and an RC 22. B 2 Sixth Order Low Pass Filter Topology Cascading Three MFB Stages 23. List of Tables, 1 Butterworth Filter Table 9, 2 Bessel Filter Table 9. 3 1 dB Chebyshev Filter Table 10, 4 3 dB Chebyshev Filter Table 10. 5 Summary of Filter Type Trade Offs 18, 6 Summary of Architecture Trade Offs 18. 1 Introduction, There are many books that provide information on popular filter types like the Butterworth. Bessel and Chebyshev filters just to name a few This paper will examine how to implement. these three types of filters, We will examine the mathematics used to transform standard filter table data into the transfer. functions required to build filter circuits Using the same method filter tables are developed that. enable the designer to go straight to the calculation of the required circuit component values. Actual filter implementation is shown for two circuit topologies the Sallen Key and the Multiple. Feedback MFB The Sallen Key circuit is sometimes referred to as a voltage controlled voltage. source or VCVS from a popular type of analysis used. It is common practice to refer to a circuit as a Butterworth filter or a Bessel filter because its. transfer function has the same coefficients as the Butterworth or the Bessel polynomial It is also. common practice to refer to the MFB or Sallen Key circuits as filters The difference is that the. Butterworth filter defines a transfer function that can be realized by many different circuit. topologies both active and passive while the MFB or Sallen Key circuit defines an architecture. or a circuit topology that can be used to realize various second order transfer functions. 2 Active Low Pass Filter Design, The choice of circuit topology depends on performance requirements The MFB is generally. preferred because it has better sensitivity to component variations and better high frequency. behavior The unity gain Sallen Key inherently has the best gain accuracy because its gain is. not dependent on component values, 2 Filter Characteristics. If an ideal low pass filter existed it would completely eliminate signals above the cutoff. frequency and perfectly pass signals below the cutoff frequency In real filters various trade offs. are made to get optimum performance for a given application. Butterworth filters are termed maximally flat magnitude response filters optimized for gain. flatness in the pass band the attenuation is 3 dB at the cutoff frequency Above the cutoff. frequency the attenuation is 20 dB decade order The transient response of a Butterworth filter. to a pulse input shows moderate overshoot and ringing. Bessel filters are optimized for maximally flat time delay or constant group delay This means. that they have linear phase response and excellent transient response to a pulse input This. comes at the expense of flatness in the pass band and rate of rolloff The cutoff frequency is. defined as the 3 dB point, Chebyshev filters are designed to have ripple in the pass band but steeper rolloff after the. cutoff frequency Cutoff frequency is defined as the frequency at which the response falls below. the ripple band For a given filter order a steeper cutoff can be achieved by allowing more. pass band ripple The transient response of a Chebyshev filter to a pulse input shows more. overshoot and ringing than a Butterworth filter, 3 Second Order Low Pass Filter Standard Form. The transfer function HLP of a second order low pass filter can be express as a function of. frequency f as shown in Equation 1 We shall use this as our standard form. FSF fc Q FSF fc, Equation 1 Second Order Low Pass Filter Standard Form. In this equation f is the frequency variable fc is the cutoff frequency FSF is the frequency. scaling factor and Q is the quality factor Equation 1 has three regions of operation below. cutoff in the area of cutoff and above cutoff For each area Equation 1 reduces to. f fc HLP f K the circuit passes signals multiplied by the gain factor K. f FSF HLP f jKQ signals are phase shifted 90 and modified by the Q factor. f fc HLP f K FSF fc signals are phase shifted 180 and attenuated by the. square of the frequency ratio, With attenuation at frequencies above fc increasing by a power of 2 the last formula describes a. second order low pass filter, Active Low Pass Filter Design 3. The frequency scaling factor FSF is used to scale the cutoff frequency of the filter so that it. follows the definitions given before, 4 Math Review. A second order polynomial using the variable s can be given in two equivalent forms the. coefficient form s2 a1s a0 or the factored form s z1 s z2 that is. P s s2 a1s a0 s z1 s z2 Where z1 and z2 are the locations in the s plane where. the polynomial is zero, The three filters being discussed here are all pole filters meaning that their transfer functions. contain all poles The polynomial which characterizes the filter s response is used as the. denominator of the filter s transfer function The polynomial s zeroes are thus the filter s poles. All even order Butterworth Bessel or Chebyshev polynomials contain complex zero pairs This. means that z1 Re Im and z2 Re Im where Re is the real part and Im is the imaginary. part A typical mathematical notation is to use z1 to indicate the conjugate zero with the positive. imaginary part and z1 to indicate the conjugate zero with the negative imaginary part Odd. order filters have a real pole in addition to the complex conjugate pairs. Some filter books provide tables of the zeros of the polynomial which describes the filter others. provide the coefficients and some provide both Since the zeroes of the polynomial are the. poles of the filter some books use the term poles Zeroes or poles are used with the factored. form of the polynomial and coefficients go with the coefficient form No matter how the. information is given conversion between the two is a routine mathematical operation. Expressing the transfer function of a filter in factored form makes it easy to quickly see the. location of the poles On the other hand a second order polynomial in coefficient form makes it. easier to correlate the transfer function with circuit components We will see this later when. examining the filter circuit topologies Therefore an engineer will typically want to use the. factored form but needs to scale and normalize the polynomial first. Looking at the coefficient form of the second order equation it is seen that when s a0 the. equation is dominated by a0 when s a0 s dominates You might think of a0 as being the. break point where the equation transitions between dominant terms To normalize and scale to. other values we divide each term by a0 and divide the s terms by c The result is. a0a1swc 1 This scales and normalizes the polynomial so that the. break point is at s a0 c, Q1 and a0 FSF the equation becomes. By making the substitutions s j2 f c 2 fc a1, FSFf 1 which is the denominator of Equation 1 our standard. fc Q FSF fc, form for low pass filters, Throughout the rest of this article the substitution s j2 f will be routinely used without. explanation, 5 Examples, The following examples illustrate how to take standard filter table information and process it into. our standard form, 4 Active Low Pass Filter Design. 5 1 Second Order Low Pass Butterworth Filter, The Butterworth polynomial requires the least amount of work because the frequency scaling. factor is always equal to one, From a filter table listing for Butterworth we can find the zeroes of the second order Butterworth. polynomial z1 0 707 j0 707 z1 0 707 j0 707 which are used with the factored form of. the polynomial Alternately we find the coefficients of the polynomial a0 1 a1 1 414 It can. be easily confirmed that s 0 707 j0 707 s 0 707 j0 707 s2 1 414s 1. To correlate with our standard form we use the coefficient form of the polynomial in the. denominator of the transfer function The realization of a second order low pass Butterworth. filter is made by a circuit with the following transfer function. Equation 2 Second Order Low Pass Butterworth Filter. This is the same as Equation 1 with FSF 1 and Q 1 0 707. 5 2 Second Order Low Pass Bessel Filter, Referring to a table listing the zeros of the second order Bessel polynomial we find. z1 1 103 j0 6368 z1 1 103 j0 6368 a table of coefficients provides a0 1 622 and a1. Again using the coefficient form lends itself to our standard form so that the realization of a. second order low pass Bessel filter is made by a circuit with the transfer function. Equation 3 Second Order Low Pass Bessel Filter From Coefficient Table. We need to normalize Equation 3 to correlate with Equation 1 Dividing through by 1 622 is. essentially scaling the gain factor K which is arbitrary and normalizing the equation. 1 274fc fc, Equation 4 Second Order Low Pass Bessel Filter Normalized Form. Equation 4 is the same as Equation 1 with FSF 1 274 and Q 1 360 1 1 274 0 577. 5 3 Second Order Low Pass Chebyshev Filter With 3 dB Ripple. Referring to a table listing for a 3 dB second order Chebyshev the zeros are given as. z1 0 3224 j0 7772 z1 0 3224 j0 7772 From a table of coefficients we get. a0 0 7080 and a1 0 6448, Active Low Pass Filter Design 5. Again using the coefficient form lends itself to a circuit implementation so that the realization of. a second order low pass Chebyshev filter with 3 dB of ripple is accomplished with a circuit. having a transfer function of the form, Equation 5 Second Order Low Pass Chebyshev Filter With 3 dB Ripple From Coefficient. Dividing top and bottom by 0 7080 is again simply scaling of the gain factor K which is. arbitrary so we normalize the equation to correlate with Equation 1 and get. 0 8414fc fc, Equation 6 Second Order Low Pass Chebyshev Filter With 3 dB Ripple Normalized Form. Equation 6 is the same as Equation 1 with FSF 0 8414 and Q 0 8414 1 0 9107 1 3050. The previous work is the first step in designing any of the filters The next step is to determine a. circuit to implement these filters, 6 Low Pass Sallen Key Architecture. Figure 1 shows the low pass Sallen Key architecture and its ideal transfer function. j2pf R1R2C1C2 j2pf R1C1 R2C1 R1C2 R4, Figure 1 Low Pass Sallen Key Architecture. At first glance the transfer function looks very different from our standard form in Equation 1 Let. us make the following substitutions K R3 R4 FSF fc. 2p R1R2C1C2, R1C1 R2C1 R1C2 1 K and they become the same. Depending on how you use the previous equations the design process can be simple or. tedious Appendix A shows simplifications that help to ease this process. 6 Active Low Pass Filter Design, 7 Low Pass Multiple Feedback MFB Architecture. Figure 2 shows the MFB filter architecture and its ideal transfer function. j2pf R2R3C1C2 j2pf R3C1 R2C1, 2 R2R3C1 1, Figure 2 Low Pass MFB Architecture. Again the transfer function looks much different than our standard form in Equation 1 Make the. following substitutions K, 2p R2R3C1C2, R3C1 R2C1 R3C1 K. and they become the same, Depending on how you use the previous equations the design process can be simple or. tedious Appendix A shows simplifications that help to ease this process. The Sallen Key and MFB circuits shown are second order low pass stages that can be used to. realize one complex pole pair in the transfer function of a low pass filter To make a Butterworth. Bessel or Chebyshev filter set the value of the corresponding circuit components to equal the. coefficients of the filter polynomials This is demonstrated later. Active Low Pass Filter Design 7, 8 Cascading Filter Stages. The concept of cascading second order filter stages to realize higher order filters is illustrated in. Figure 3 The filter is broken into complex conjugate pole pairs that can be realized by either. Sallen Key or MFB circuits or a combination To implement an n order filter n 2 stages are. required Figure 4 extends the concept to odd order filters by adding a first order real pole. Theoretically the order of the stages makes no difference but to help avoid saturation the. stages are normally arranged with the lowest Q near the input and the highest Q near the. output Appendix B shows detailed circuit examples using cascaded stages for higher order. Complex Conjugate Pole Pairs, Input Output, VI Stage 1 Stage 2 Stage n 2 VO. Buffer Buffer, Optional Lowest Q Highest Q Optional. Figure 3 Building Even Order Filters by Cascading Second Order Stages. Active Low Pass Filter Design Jim Karki AAP Precision Analog ABSTRACT This report focuses on active low pass filter design using operational amplifiers Low pass filters are commonly used to implement antialias filters in data acquisition systems Design of second order filters is the main topic of consideration Filter tables are developed to simplify circuit design based on the idea of

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