Fundamentals of Active Filters

Transcription

1 Fundamentals of Active Filters This training module covers active filters. It introduces the three main filter optimizations, which include: Butterworth, Chebyshev and Bessel. The general transfer function is presented without the tedious mathematical derivations. This training module gives a step-by-step method of how to select the best operational amplifier(op-amp) for the active filter circuit and how to calculate the individual circuit components. Course Map/Table of Contents 1. Course Navigation Course Navigation 2. Active Filter Fundamentals What is a Filter 2.2 Types of Filters 2.3 Passive vs. Active Filters 3. Low Pass Filters Basic Ideal Filter Response 3.2 Butterworth Low-pass Filter Response 3.3 Chebyshev Low-pass Filter Response 3.4 Bessel Low-pass Filter Response 3.5 First Order Active Filters 3.6 Selecting the Passive Components 3.7 The "Scalable" Filter 3.8 Second Order Low-pass Filter 3.9 Multiple Feedback Topology (MFB) 3.10 Higher Order Filters 3.11 Component Selection 3.12 Reference Material 4. High Pass Filters High-Pass Filter Design 4.2 Second Order High Pass Filter Course Navigation 1.1 Course Navigation Course Navigation This course is organized like a book with multiple chapters. Each chapter may have one or more pages. The previous and next arrows move you forward and back through the course page by page. The left navigation bar takes you to any chapter. It also contains the bookmarking buttons, 'save' and 'go to.' To save your place in a course, press the 'save' button. The next time you open the course, clicking on 'go to' will take you to the page you saved or bookmarked. The top services bar contains additional information such as glossary of terms, who to go to for help with this subject and an FAQ. Clicking home on this bar will take you back to the course beginning. Don't miss the hints, references, exercises and quizzes which appear at the bottom of some pages. Active Filter Fundamentals

2 This chapter will cover the types of filters and discuss the difference between passive and active filters. 2.1 What is a Filter 2.2 Types of Filters 2.3 Passive vs. Active Filters What is a Filter Webster dictionary defines a filter as a device that passes electronic signals at certain frequencies or frequency ranges while preventing the passage of others. Filter circuits are used in a wide variety of applications. The following are a few examples: Modems and speech processing use band pass filters in the audio frequency range (0 khz to 20 khz). Audio circuits use filters for bass and treble control. Telephone central offices use high frequency bandpass filters (several hundred MHz) for channel selection. System power supplies use band rejection filters to suppress the 60 Hz line frequency. Anti-aliasing low-pass filters, as well as low-pass noise filters, are used in the signal conditioning stage. Data acquisition systems. Types of Filters There are five basic filter types. Four of them can be included in one category and the fifth is its own type. They are as follows: Frequency selective circuits. Low-pass filters ideally pass all frequencies within the bandpass and reject frequencies outside the band. High-pass filters ideally have a pass band between a low and high cut off frequency and reject frequencies outside the band. Band-pass filters ideally allow a narrow band of frequencies to pass and reject all others. Notch filters ideally reject only a specific, and often very narrow, band of frequencies and pass all others. Time-delay filters or all-pass filters pass all frequencies equally in amplitude but change the phase of the input signals depending upon their frequency.

3 You can find band-pass filters in familiar places, such as in the tuning circuitry of a radio, which allows the user to select a particular station and block out all others. Passive vs. Active Filters Passive filters consist of only passive elements which include resistors, inductors and capacitors. On the other hand active filters consist of passive elements along with active devices, such as transistors or op-amps, to obtain a response equal to or better than conventional filters. The advantages of active filters over passive filters are: No insertion loss. The op-amp can provide gain if needed. Active filtering practically eliminates insertion loss due to the high input impedance and low output impedance of an op-amp. Furthermore, with active filtering, it is possible to attenuate unwanted frequencies while amplifying desired frequencies. Cost. Active filter components are more economical than inductors. Inductors are typically bulky, costly and depart further from ideal models compared to capacitors. Inductors do not lend themselves to IC-type mass production. Tuning. Active filters are easily tuned and adjusted over a wide range without altering the desired response. Isolation. Active filters have good isolation due to their high input impedance and low output impedance. This assures minimal interaction between the filter and its load. At High frequencies of greater than 10MHz, the filters usually consist of passive components such as inductors (L), resistors (R), and capacitors (C). They are called LRC filters.

4 Low Pass Filters This chapter covers the low-pass filter response of the three main types of filters. Links to tables are included in the chapter so that values of the filter components can be calculated and chosen based on availability for the required application. 3.1 Basic Ideal Filter Response 3.2 Butterworth Low-pass Filter Response 3.3 Chebyshev Low-pass Filter Response 3.4 Bessel Low-pass Filter Response 3.5 First Order Active Filters 3.6 Selecting the Passive Components 3.7 The "Scalable" Filter 3.8 Second Order Low-pass Filter 3.9 Multiple Feedback Topology (MFB) 3.10 Higher Order Filters 3.11 Component Selection 3.12 Reference Material Basic Ideal Filter Response Filters are based on the approximation of the ideal response characteristic. The ideal low-pass response is characterized by the frequency called the cutoff frequency(fc). As long as the input signal frequency is less than the fc, the signal passes through the filter with unchanged amplitude, while signals with frequencies higher than fc undergo complete attenuation. A common low-pass filter application is the removal of high frequency noise from a signal. Most often the passband is considered to extend to the -3dB point. The cutoff frequency, fc, is the end of the passband.

5 In comparison to ideal low-pass filtering, actual filter response lacks the following characteristics: The passband gain varies long before fc. The transition from the passband into the stopband is not sharp. The phase response is not linear, thus the amount of signal distortion is increased. One application for the low-pass filter is in a light sensing instrument using a photodiode. When the light levels are low, the output of the photodiode can be very small, causing it to be partially obscured by the noise of the sensor and its amplifier, whose spectrum can extend to very high frequencies. If a low-pass filter, with a high cutoff frequency, is placed at the output of the amplifier, it will allow the desired signal frequencies to pass and this will reduce the overall noise level. Butterworth Low-pass Filter Response The three predominant filter optimization are Butterworth, Chebyshev, and Bessel. Butterworth is one of the most commonly used filter optimizations. It has the following characteristics: It provides monotonic response, the maximally flat passband response. For this reason, it is sometimes called a flat-flat filter. Butterworth filters are used as anti-aliasing filters in data converter applications where precise signal levels are required across the entire passband. The frequency response below shows the transition band ( fc to f1) where the response shifts from the passband to the stopband. The passband ends when A 1 is -3db down from the low frequency response A 0. The stopband enters when the response drops to some predetermined value A 2. The Butterworth filter has poor phase response characteristics. Increasing the order of the filter flattens the passband response and steepens the stopband falloff.

6 the passband response and steepens the stopband falloff. Chebyshev Low-pass Filter Response The Chebyshev low-pass filter has the following characteristics: It provides a higher gain rolloff above fc. (The steepest transition from passband to stopband.) Passband gain is not monotone, but contains ripples of constant magnitude. For a given filter order, the higher the passband ripples, the higher the filter's rolloff. Filters with even order numbers generate ripples above the 0-dB line, while filters with odd order numbers create ripples below 0 db. Chebyshev filters are often used in filter banks, where the frequency content of a signal is of more importance than the constant amplification. Bessel Low-pass Filter Response The Bessel low-pass filter has a linear phase response over a wide frequency range, which results in a constant group delay in that frequency range; therefore, it provides an optimum square wave transmission. Filters characterized by a flat amplitude response may have large phase shifts. The result is that a signal in the passband will suffer distortion of its waveform. The characteristics of the Bessel filter are as follows: It has a linear phase response over a wide frequency range. (The waveform is not distorted.) Passband gain is not as flat as the Butterworth low-pass, and the transition from passband to stopband is not as sharp as the Chebyshev low-pass filter.

7 A Quick Filter Quiz 1 A customer is designing an anti-aliasing filter in the data converter application, which filter optimization would you recommend? 1. Butterworth 2. Chebyshev 3. Bessel 4. LRC filter 1 Answer: Butterworth 2 A customer's application requires a low-pass filter of 10 MHz, which filter optimization would you recommend? 1. Butterworth 2. Chebyshev 3. Bessel 4. LRC filter 2 Answer: LRC filter First Order Active Filters The simplest active filter is obtained from the basic op-amp configurations by using a capacitor as one of its external components. First order active filters should be limited to applications that require only a low Quality factor, Q( 100), low frequency ( 5kHz) and low gain ( 10).The op amp selected for these filters should have an open loop voltage gain at the highest frequency of interest at least 50 times larger than the gain of the filter at that frequncy. The slew rate (SR) of the amplifier should be determined using the following equation: SR π*fc* Voutp-p * 10e-6 V/usec The transfer function of first order active filter is where H 0 is the DC gain, ω = 2πf, and a 1 is the filter coefficient. The value of a1 can be found in the coefficient tables. Placing a capacitor in parallel with the feedback resistors of an inverting configuration converts the circuit into a low-pass filter with gain.

8 The transfer function of this circuit is Similarly, for a non-inverting configuration, adding a capacitor at the input of the op-amp converts the circuit into a non-inverting low pass filter. The transfer function here is What if you put many filters in series? If the design requirement is a low-pass butterworth filter with flat passband and sharp transition to the stop band, then the ultimate rate of falloff will always be 20ndB/decade, where n is the number of poles. You need one capacitor for each pole, so the required ultimate rate of falloff of filter response determines, roughly, the complexity of the filter. If you decide to use a 6 pole low-pass filter, you are guaranteed an ultimate rolloff of 120db/decade at high At low frequency. frequencies a capacitor tends to behave as an open circuit compared with the surrounding elements and at high frequencies it tends to behave as a short circuit. A Quick Filter Quiz When selecting an op amp for a first order low-pass filter design, what is the minimum SR required for fc = 5 khz, V of outp-p 20V? V/msec V/µsec V/µsec V/msec 1 Answer: V/µsec Selecting the Passive Components The quality of the passive components is very important in the performance of an active filter. Carbon composition resistors should not be used in critical filters and are useful mainly for room temperature breadboarding of initial concepts. Ceramic capacitors are generally not suitable for active filter applications because of their relatively poor performance. The NPO ceramic capacitors are an exception. They are excellent for values smaller than 2000 pf. To minimize the variations of F c and Q, NPO (COG) ceramic capacitors are recommended for high-performance filters. These capacitors hold their nominal value over a wide temperature and voltage range. The various temperature characteristics of ceramic capacitors are identified by a threesymbol code such as: COG, X7R, Z5U, and Y5V. COG-type ceramic capacitors are the most precise. Their nominal values range from 0.5 pf to approximately 47 nf with initial tolerence from ± 0.25% for smaller values and up to ± 1% for higher values. Their capacitance drift over temperature is typically 30ppm/Cº.

9 X7R-type ceramic capacitors range from 100 pf to 2.2 µf with an initial tolerance fo +1% and a capacitance drift over temperature of ± 15%. For higher values, tantalum electrolytic capacitors should be used. Other precision capacitors are silver mica, metallized polycarbonate, and for high temperatures, polypropylene or polystyrene. Mylar capacitors are commonly used, but polypropylene (or polystyrene) or even teflon capacitors are often used in high performance filters. Capacitor values for active filter designs should be selected first and standard values should be used. A wider range of standard resistor values is generally available, and this higher resolution of values can then be used to obtain the desired filter response. Now that you know what passive components to choose, here is how to select the values for the components. From the first order non-inverting low-pass filter transfer function you can solve for R 1 and R 2. From the first order inverting low-pass filter transfer function you can solve for R 1 and R 2. The value for a 1 is taken from one of the coefficient tables below. Note that all filter types are identical in their first order and a 1 = 1 The following tables contain the coefficients for the three filter types: Bessel, Butterworth and Chebyshev.The table headers consist of the following quantities: n is the filter order. i is the number of the partial filter. ai and bi are the filter coefficients. ki is the ratio of the corner frequency of a partial filter, fci, to the corner frequency of the overall filter, fc. Qi is the quality factor of the partial filter. Since capacitor values are not as finely subdivided as resistor values, the capacitor values should be defined prior to selecting resistors. Filter Coefficient Table - Butterworth Filter Coefficient Table - Bessel Filter Coefficient Table - Chebyshev for 0.5 db passband ripple Filter Coefficient Table - Chebyshev for 1 db passband ripple Filter Coefficient Table - Chebyshev for 2 db passband ripple For a first order inverting configuration, find the value of the external components to achieve a -3dB frequency of 1kHz with a dc gain of 20 db? DC gain of 20 db = 10V/V = R 2 /R 1 R 2 = 10R 1 also R 2 = 1/(2πfcC1 Use a value for C1 that is readily available. Results: C 1 = 1nF

10 C 1 = 1nF R 2 = 158 kω R 1 = 15.8 kω The "Scalable" Filter Once a filter is designed, there may be a need to change the cutoff frequency. The procedure used to convert an original cutoff frequency to a new cutoff frequency is called frequency scaling. To change the cutoff frequency, multiply R or C, but not both, by the ratio of the original cutoff frequency to the new cutoff frequency. CAUTION! In an actual circuit, if the resistor values become too small, excessive loading may be placed on the output of the amplifier. This will reduce accuracy and can exceed either the output current or the power disspation capabilites of the amplifier. You are asked to change the cutoff frequency of a low pass filter from 1kHz to 1.6Khz. The original design has a capacitor value of 0.01 uf and a resistor value of 115.9K ohms. The new resistor value is kω * ( 1/1.6) = 72.4 kω A Quick Filter Quiz 1 Design a low-pass filter at a cutoff frequency of 1 khz with a passband gain of 2, assume C = 0.01µF. 1. R1 = 1/(10 3 )(10-8 ) = 100 kω. Since the passband gain is 2, the gain resistors must be equal. Therefore, let R 2 = R 3 = 10kΩ. 2. R1 = 1/(10 3 )(10-8 ) = 100 kω. Since the passband gain is 2, the gain resistors can't be equal. Therefore, let R 2 = 10 kω, and R 3 = 5 kω 3. R1 = 1/(2π)(10 3 )(10-8 ) = 15.9 kω. Since the passband gain is 2, the gain resistors must be equal. Therefore, let R 2 = R 3 = 10kΩ. 4. R1 = 1/(2π)(10 3 )(10-8 ) = 15.9 kω. Since the passband gain is 2, the gain resistors can't be equal. Therefore, let R 2 = 10 kω, and R 3 = 5 kω 1 Answer: 3. R 1 = 1/(2π)(10 3 )(10-8 ) = 15.9 kω. Since the passband gain is 2, the gain resistors must be equal. Therefore, let R 2 = R 3 = 10kΩ. 2 Using the frequency scaling technique, convert the 1 khz frequency of the low pass filter. 1. To change the cutoff frequency from 1 khz to 1.6 khz, multiply the 15.9 kω by (original cutoff frequency/new cutoff frequency) = 0.625, new resistor value = 9.94 Ω. 2. To change the cutoff frequency from 1 khz to 1.6 khz, mulitply the 15.9 kω by (new cutoff frequency/original frequency) =1.6, new resistor value = 25.44Ω. 2 Answer: 1. To change the cutoff frequency from 1 khz to 1.6 khz, multiply the 15.9 kω by (original cutoff frequency/new cutoff frequency) = 0.625, new resistor value = 9.94 Ω. Second Order Low-pass Filter Second order filters are important because higher order filters can be designed using them. There are two topologies for a second order low pass filter, the Sallen-Key and the Multiple Feedback ( MFB) topology. The second order response provides a high frequency asymptotic slope twice as steep as the first order. The unity gain topology of the Sallen-Key is usually applied in filter designs with high gain accuracy, unity gain, and low Q (less than 3).

11 The transfer function of the general Sallen-Key Low-pass filter is For a unity gain circuit the transfer function simplifies to The coefficients of the transfer function are In order to obtain real values under the square root, C 2 must satisfy the following condition Given C 1 and C 2, the resistor values R 1 and R 2 are calculated through In actual applications, Q may range from as low as 0.5 to as high as 100, with values near unity being by far the most common. Second Order Unity Gain Chebyshev Low Pass Filter Design a second order unity gain Chebyshev low pass filter with a corner frequency Fc= 3 KHz and a 3-dB passband ripple. From the table (the Chebyshev coefficients for 3-dB ripple) obtain the coefficients a and b for the second order filter with a = and b = Specifying C as 22 nf. 1 Calculate the values of C, R and R Results C = 150 nf R = 1.26 KΩ R = 1.30KΩ Multiple Feedback Topology (MFB) The MFB topology is commonly used in filters that have high Qs and require a high gain.

12 The transfer function of the circuit is as follows The coefficients are Given C 1 and C 2 and solving for the resistors R 1 - R 3 In order to obtain values for R 2, C 2 must satisfy the following condition Higher Order Filters We know that the gain of the filter changes at the rate of 20dB/decade for first order filters and at 40dB/decade for second order filters. This means that, as the order of the filter is increased, the actual response of the filter approaches its ideal characteristics. Higher order filters, such as third, fourth, fifth and so on, are formed simply by using the first and second order filters. For example, a third order low-pass filter is formed by connecting a first and second order low pass filter in series. A fourth order low-pass filter is composed of two cascaded second order low-pass sections. The overall gain of the filter is equal to the product of the individual voltage gain's of the filter sections. Step by Step Design Example: Design a fifth order unity gain Butterworth filter with the corner frequency f C = 50 khz. First obtain the coefficients for a fifth order Butterworth filter from the tables. First filter is The value of R 1 is calculated using this equcation. The closest 1% value is 3.16 kω Second Filter The calculation of the third filter is identical to the calculation of the second filter, except that a and b are replaced by a and b, resulting in different capacitor and resistor values. specify C as 330pF and obtain C with 1 2 C2 = 3.46 nf, the closest 10% capacitor value is 4.7 nf. With these capacitor values, the values for R1 and R2 are R 1 = 1.45 kω with the closest 1% value being 1.47 kω R 2 = 4.51 kω with the closest 1% value being 4.53 kω Here is the final fifth order unity gain Butterworth Low Pass Filter

13 National Semiconductor's WEBENCH Online Design Environment has been selected from hundreds of entries as the winner of EDN Magazine's 2004 Innovator/Innovation Awards. EDN selected National Semiconductor's WEBENCH Active Filter Designer as the winner in the EDA (electronic design automation) category for its ability to speed the creation of sophisticated filters for data acquisition and signal conditioning applications. Designed from the ground up, the online tool lets engineers design advanced, highly customized filters for all standard filter types: low pass, high pass, band pass and band stop. Press Release March 9, WEBENCH Design the fifth order filter using WEBENCH Specify the design using filter approximation. Compare WEBENCH results to the calculations above. Component Selection To minimize variations of fc and Q, NPO ceramic capacitors are recommended for high performance filters. These capacitors hold their nominal value over a wide temperature and voltage range. For high performance filters, 0.1% resistors are recommended.resistor values should stay within the range of 1kΩ to 100kΩ. The lower limit avoids excessive current draw from the op amp output, which is important in power sensitive applications. The upper limit of 100KΩ is to avoid excessive resistor noise. Op Amp Selection The most important op amp parameter for proper filter functionality is the unity gain bandwidth. In general, the open loop gain should be 100 times ( 40 db above) the peak gain of a filter section to allow a maximum gain error of 1%. Use an op amp with adequate bandwidth, where the filter 3db > 10 times the highest frequency in passband of the filter. Other op amp performance required low noise and low signal distortion. The slew rate is SR = π * Vpp * fc. Example for a 100kHz filter with 5Vpp the output requires an SR = 1.57V/µs The response of the filter in the lab will deviate from the reponse predicated by theory. This is due to the component tolerance and the op-amp nonidealities. Even if some of the components are made adjustable to allow for fine tuning, deviations will still arise because of the component aging and thermal drift. Reference Material OA-27 Low-sensitivity, Lowpass Filter Design AN-779 A Basic Introduction to Filters - Active, Passive, and Switched-Capacitor Intuitive IC Op Amps from Basics to Useful Applications by Thomas M Frederiksen High Pass Filters This chapter will cover high-pass filters response of the two topologies: Sallen-Key and Multiple Feedback. High-Pass Filter Design

14 High-Pass Filter Design A high-pass filter is easily created from a low-pass filter by replacing the resistors of the low-pass filter with capacitors, and its capacitors with resistors. First Order Non-inverting High-Pass Filter The transfer function of the circuit is First order Inverting High-Pass Filter The transfer function of the circuit is The coefficient a 1 is the same for both inverting and non-inverting circuits To calculate the resistor value of R 1, specify the corner frequency (fc), the dc gain (Adc)and capacitor (C 1 ) The resistor value R 2, differs for the inverting and non-inverting circuits. For the non-inverting circuit For the inverting circuit

15 Second Order High Pass Filter High-Pass filters use the same two topologies as the Low-Pass filters: Sallen-Key and Multiple Feedback. The only difference is that the positions of the resistors and the capacitors have changed. Sallen-key Topology For low Q filters with high gain accuracy, the unity gain Sallen-Key topology is applied. To simplify the circuit design, choose unity gain, and C 1 =C 2 =C. The transfer function of the circuit is Given C, the resistor values for R 1 and R 2 are calculated through Multiple Feedback Topology (MFB)

16 The MFB topology is commonly used in filters that have high Qs and require high gain. Given Capacitors C and C 2 and solving for resistors R 1 and R 2 : Similar to low-pass filters, higher-order high-pass filters are designed by cascading first-order and second-order filter stages. The filter coefficients are the same ones used for the low-pass filter design and are listed in the coefficient tables found in the Low-Pass filter section. The passband gain of a MFB high-pass filter can vary significantly due to the wide tolerances of the two capacitors C and C 2. To keep the gain variation at a minimum, it is necessary to use capacitors with tight tolerance values. Design a third order unity gain Bessel high-pass filter with the corner frequency f c =1kHz. The coefficients for the third order Bessel filter from the tables are Filter 1 a 1 =0.756, b 1 =0 Filter 2 a 2 =0.9996, b 2 = First Filter C 1 = 100nF, Calculate R 1 = 2.105kΩ, the closest 1% value is 2.1KΩ. Second Filter C=100nF, calculate R 1 =3.18kΩ, the closest 1% value is 3.16kΩ. Calculate R 2 = 1.67kΩ, closest 1% value is 1.65kΩ. Maximum Gain The Maximum gain of a fitler is given by H O. It is the ratio of V O to V i. Decibels (db) are often used as a relative measure of a filter's gain. Group delay Group delay is a measure of how long it takes a signal to traverse a network, or its transmit time. It is a strong function of the length of the netwrk, and usually a weak function of frequency. It is expressed in units of time, pico-seconds for short distances or nanoseconds for longer distances. Quality Factor The quality factor, Q, is a dimensionless number used to measure the selectivity of a filter and is expressed as the ratio of the filter's center frequency to the bandwidth. For example, given a filter with a fixed center frequency, decreasing the filter's bandwidth (i.e. increasing its sharpness) increases Q.

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