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CHAPTER 4: ACTIVE FILTERS

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O BJECTIVES : Describe three types of filter response characteristics and other parameters. Describe and analyze the gain-versus-frequency responses of basic types of filters. Identify and analyze active low-pass filters. Identify and analyze active high-pass filters. Analyze basic types of active band-pass filters. Describe basic types of active band-stop filters.

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Filter passing rejectingattenuating selectivity Filter is a circuit used for signal processing due to its capability of passing signals with certain selected frequencies and rejecting or attenuating signals with other frequencies. This property is called selectivity. Filter can be passive or active filter. Passive filters Passive filters : The circuits built using RC, RL, or RLC circuits. Active filters Active filters : The circuits that employ one or more op- amps in the design an addition to resistors and capacitors.

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Filter Filter is a circuit that passes certain frequencies and rejects all others. The passband is the range of frequencies allowed through the filter. The critical frequency defines the end (or ends) of the passband.

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Fig. 1-1: Low-pass filter responses VoVo low-pass filter A low-pass filter is one that passes frequency from dc to fc and significantly attenuates all other frequencies. The simplest low-pass filter is a passive RC circuit with the output taken across C.

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Passband Passband of a filter is the range of frequencies that are allowed to pass through the filter with minimum attenuation (usually defined as less than -3 dB of attenuation). Transition region Transition region shows the area where the fall-off occurs. Stopband Stopband is the range of frequencies that have the most attenuation. Critical frequencyf c Critical frequency, f c, (also called the cutoff frequency) defines the end of the passband and normally specified at the point where the response drops – 3 dB (70.7%) from the passband response.

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At low frequencies, X C is very high and the capacitor circuit can be considered as open circuit. Under this condition, V o = V in or A V = 1 (unity). At very high frequencies, X C is very low and the V o is small as compared with V in. Hence the gain falls and drops off gradually as the frequency is increased. VoVo V in

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bandwidthidealf c The bandwidth of an ideal low-pass filter is equal to f c : X C = R When X C = R, the critical frequency of a low-pass RC filter can be calculated using the formula below: (4-1) (4-2)

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high-pass filter A high-pass filter is one that significantly attenuates or rejects all frequencies below f c and passes all frequencies above f c. The simplest low-pass filter is a passive RC circuit with the output taken across R. VoVo Fig. 1-2: High-pass filter responses

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X C = R The critical frequency for the high pass-filter also occurs when X C = R, where (4-3)

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band-pass filter lower-frequency limitupper-frequency limit A band-pass filter passes all signals lying within a band between a lower-frequency limit and upper-frequency limit and essentially rejects all other frequencies that are outside this specified band. The simplest band-pass filter is an RLC circuit. Fig. 1-3: General band-pass response curve.

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bandwidth (BW) difference upper critical frequency (f c2 ) lower critical frequency (f c1 ) The bandwidth (BW) is defined as the difference between the upper critical frequency (f c2 ) and the lower critical frequency (f c1 ). center frequencyf o The frequency about which the passband is centered is called the center frequency, f o, defined as the geometric mean of the critical frequencies. (4-4) (4-5)

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quality factor (Q) The quality factor (Q) of a band-pass filter is the ratio of the center frequency to the bandwidth. The quality factor (Q) can also be expressed in terms of the damping factor (DF) of the filter as (4-6) (4-7)

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Band-stop filteropposite within rejectedoutsidepassed Band-stop filter is a filter which its operation is opposite to that of the band-pass filter because the frequencies within the bandwidth are rejected, and the frequencies outside bandwidth are passed. Its also known as notch, band-reject or band-elimination filter Fig. 1-4: General band-stop filter response.

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E XAMPLE 1 A certain band-pass filter has a center frequency of 15 kHz and a bandwidth of I kHz. Determine the Q and classify the filter as narrow-band or wide-band. Answer: Q = 15

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Butterworth ChebyshevBessel The characteristics of filter response can be Butterworth, Chebyshev, or Bessel characteristic. Fig. 1-5: Comparative plots of three types of filter response characteristics. Butterworth characteristic Filter response is characterized by flat amplitude response flat amplitude response in the passband. Provides a roll-off rate of -20 dB/decade/pole. Filters with the Butterworth response are normally used when all frequencies in the passband same gain must have the same gain.

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Chebyshev characteristic Filter response is characterized by overshootripples overshoot or ripples in the passband. Provides a roll-off rate greater than -20 dB/decade/pole. Filters with the Chebyshev response can be implemented with fewer polesless complex fewer poles and less complex circuitry circuitry for a given roll-off rate. Bessel characteristic linear characteristic Filter response is characterized by a linear characteristic, meaning that the phase shift increases linearly with frequency. Filters with the Bessel response are used for filtering pulse waveforms without distorting the shape of waveform. Bessel characteristic linear characteristic Filter response is characterized by a linear characteristic, meaning that the phase shift increases linearly with frequency. Filters with the Bessel response are used for filtering pulse waveforms without distorting the shape of waveform.

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damping factor (DF) The damping factor (DF) primarily determines if the filter will have a Butterworth, Chebyshev, or Bessel response. an amplifiera negative feedback circuitRC circuit This active filter consists of an amplifier, a negative feedback circuit and RC circuit. non- inverting configuration The amplifier and feedback are connected in a non- inverting configuration. DF is determined by the negative feedback and defined as Fig. 1-6: Diagram of an active filter. (4-8)

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Parameter for Butterworth filters up to four poles are given in the following table. Notice that the gain is 1 more than this resistor ratio. For example, the gain implied by the the ratio is (4.0dB).

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V ALUES FOR THE B UTTERWORTH RESPONSE

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E XAMPLE 2 If resistor R 2 in the feedback circuit of an active single-pole filter of the type in figure below is 10kΩ, what value must R 1 be to obtain a maximally flat Butterworth response? Answer: R1 = 5.86kΩ

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critical frequencyf c The critical frequency, f c is determined by the values of R and C in the frequency-selective RC circuit. For a single-pole (first-order) filter, the critical frequency is The above formula can be used for both low-pass and high-pass filters. Fig. 1-7: One-pole (first-order) low-pass filter. (4-9)

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The number of poles determines the roll-off rate of the filter. For example, a Butterworth response produces -20 dB/decade/pole. This means that: one-pole (first-order) one-pole (first-order) filter has a roll-off of -20 dB/decade; two-pole (second-order) two-pole (second-order) filter has a roll-off of -40 dB/decade; three-pole (third-order) three-pole (third-order) filter has a roll-off of -60 dB/decade; and so on.

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cascading The number of filter poles can be increased by cascading. To obtain a filter with three poles, cascade a two-pole and one-pole filters. Fig. 1-8: Three-pole (third-order) low-pass filter.

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Advantages of active filters over passive filters (R, L, and C elements only): no signal attenuation 1.By containing the op-amp, active filters can be designed to provide required gain, and hence no signal attenuation as the signal passes through the filter. No loading problem 2. No loading problem, due to the high input impedance of the op-amp prevents excessive loading of the driving source, and the low output impedance of the op-amp prevents the filter from being affected by the load that it is driving. Easy to adjust over a wide frequency range 3. Easy to adjust over a wide frequency range without altering the desired response.

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Fig. 1-9: Single-pole active low-pass filter and response curve. This filter provides a roll-off rate of -20 dB/decade above the critical frequency.

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The close-loop voltage gain is set by the values of R 1 and R 2, so that (4-10)

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Sallen-Key is one of the most common configurations for a two-pole filter. It is also known as a VCVS (voltage-controlled voltage source) filter. Fig. 1-10: Basic Sallen-Key low-pass filter. There are two low-pass RC circuits that provide a roll- off of -40 dB/decade above f c (assuming a Butterworth characteristics). One RC circuit consists of R A and C A, and the second circuit consists of R B and C B.

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The critical frequency for the Sallen-Key filter is (4-11) If R A = R B = R and C A = C B = C, the critical frequency can be expressed as:

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E XAMPLE 3 Determine the critical frequency of the Sallen-Key low- pass filter in Figure below, and set the value of R 1 for an appropriate Butterworth response. Answer: f c = 7.23 kHz, R 1 = 586Ω

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A three-pole filter is required to provide a roll-off rate of -60 dB/decade. This is done by cascading a two-pole Sallen-Key low-pass filter and a single-pole low-pass filter. Fig. 1-11: Cascaded low-pass filter: third-order configuration.

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Four-pole filter is obtained by cascading Sallen-Key (2-pole) filters. Fig. 1-12: Cascaded low-pass filter: fourth-order configuration.

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Example 4 Determine the cutoff frequency, the pass-band gain in dB, and the gain at the cutoff frequency for the active filter of Fig. 1-7 with C = μF, R = 3.3 kΩ, R 1 = 24 kΩ, and R 2 = 2.2 kΩ Fig. 1-7: One-pole (first-order) low-pass filter.

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In high-pass filters, the roles of the capacitor and resistor are reversed in the RC circuits. The negative feedback circuit is the same as for the low- pass filters. Fig. 1-13: Single-pole active high-pass filter and response curve.

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Components R A, C A, R B, and C B form the two-pole frequency- selective circuit. The position of the resistors and capacitors in the frequency- selective circuit. The response characteristics can be optimized by proper selection of the feedback resistors, R 1 and R 2. Fig. 1-14: Basic Sallen-Key high-pass filter.

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As with the low-pass filter, first- and second-order high-pass filters can be cascaded to provide three or more poles and thereby create faster roll-off rates. Fig. 1-15: A six-pole high-pass filter consisting of three Sallen-Key two-pole stages with the roll-off rate of -120 dB/decade.

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Example 5 Determine the cutoff frequency, the pass-band gain in dB, and the gain at the cutoff frequency for the active filter of Fig. 5-7 with C = 0.02 μF, R = 5.1 kΩ, R 1 = 36 kΩ, and R 2 = 3.3 kΩ Fig. 1-7: One-pole (first-order) high-pass filter.

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Filters that build up an active band-pass filter consist of a Sallen-Key High- Pass filter and a Sallen-Key Low-Pass filter. Fig. 1-16: Band-pass filter formed by cascading a two-pole high-pass and a two-pole low-pass filters.

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Both filters provide the roll-off rates of –40 dB/decade, indicated in Fig The critical frequency of the high-pass filter, f C1 must be lower than that of the low-pass filter, f C2 to make the center frequency overlaps. Fig. 1-17: The composite response curve of a high-pass filter and a low- pass filter.

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The lower frequency, f c1 of the pass-band is calculated as follows: The upper frequency, f c2 of the pass-band is determined as follows: The center frequency, f o of the pass-band is calculated as follows: (4-12) (4-13) (4-14)

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Multiple-feedback band-pass filter is another type of filter configuration. The feedback paths of the filter are through R 2 and C 1. R 1 C 1 low- pass filterR 2 C 2 high-pass filter R 1 and C 1 provide the low- pass filter, and R 2 and C 2 provide the high-pass filter. The center frequency is given as Fig. 1-18: Multiple-feedback band-pass filter. (5-15)

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For C 1 = C 2 = C, the resistor values can be obtained using the following formulas: (4-16) (4-17) (4-18)

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E XAMPLE 6 Determine the center frequency, maximum gain and bandwidth for the filter in figure below. Answer: fo = 736 Hz, Ao = 1.32, BW = 177 Hz

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State-variable filtera summing amplifiertwo op- amp integrators State-variable filter contains a summing amplifier and two op- amp integrators that are combined in a cascaded arrangement to form a second-order filter. Besides the band-pass (BP) output, it also provides low-pass (LP) and high-pass (HP) outputs. Fig. 1-19: State-variable filter.

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Fig. 1-20: General state-variable response curve.

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bandwidthdependent quality factorQindependent In the state-variable filter, the bandwidth is dependent on the critical frequency and the quality factor, Q is independent on the critical frequency. The Q is set by the feedback resistors R 5 and R 6 as follows: (4-19)

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Biquad filteran integratoran inverting amplifieran integrator Biquad filter contains an integrator, followed by an inverting amplifier, and then an integrator. bandwidthindependentQ dependent In a biquad filter, the bandwidth is independent and the Q is dependent on the critical frequency. Fig. 1-21: A biquad filter.

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Band-stop filters reject a specified band of frequencies and pass all others. The response are opposite to that of a band-pass filter. notch filters Band-stop filters are sometimes referred to as notch filters. R 3 has been moved and R 4 has been added This filter is similar to the band-pass filter in Fig except that R 3 has been moved and R 4 has been added. Fig. 1-22: Multiple-feedback band-stop filter.

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Summing the low-pass and the high-pass responses of the state- variable filter with a summing amplifier creates a state variable band-stop filter. Fig. 1-23: State-variable band-stop filter.

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~ E ND OF CHAPTER 4 ~

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