# CHAPTER 4: ACTIVE FILTERS.

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CHAPTER 4: ACTIVE FILTERS

Objectives: 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.

INTRODUCTION 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: The circuits built using RC, RL, or RLC circuits. Active filters : The circuits that employ one or more op- amps in the design an addition to resistors and capacitors.

Basic filter Responses
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.

BASIC FILTER RESPONSES
Basic 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. Vo Fig. 1-1: Low-pass filter responses

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 shows the area where the fall-off occurs. Stopband is the range of frequencies that have the most attenuation. Critical frequency, fc, (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.

Vin Vo At low frequencies, XC is very high and the capacitor circuit can be considered as open circuit. Under this condition, Vo = Vin or AV = 1 (unity). At very high frequencies, XC is very low and the Vo is small as compared with Vin. Hence the gain falls and drops off gradually as the frequency is increased.

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

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

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

Fig. 1-3: General band-pass response curve.
Basic Band-Pass Filter 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.

The bandwidth (BW) is defined as the difference between the upper critical frequency (fc2) and the lower critical frequency (fc1). (4-4) The frequency about which the passband is centered is called the center frequency, fo, defined as the geometric mean of the critical frequencies. (4-5)

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

Fig. 1-4: General band-stop filter response.
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.

Example 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

FILTER RESPONSE CHARACTERISTICS
The characteristics of filter response can be Butterworth, Chebyshev, or Bessel characteristic. Butterworth characteristic Filter response is characterized by 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 must have the same gain. Fig. 1-5: Comparative plots of three types of filter response characteristics.

Chebyshev characteristic
Filter response is characterized by 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 poles and less complex circuitry for a given roll-off rate. Bessel 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.

Damping Factor The damping factor (DF) primarily determines if the filter will have a Butterworth, Chebyshev, or Bessel response. This active filter consists of an amplifier, a negative feedback circuit and RC circuit. 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)

Damping Factor 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).

Values for the Butterworth response

Example 2 If resistor R2 in the feedback circuit of an active single-pole filter of the type in figure below is 10kΩ, what value must R1 be to obtain a maximally flat Butterworth response? Answer: R1 = 5.86kΩ

Critical Frequency and Roll-Off Rate
The critical frequency, fc 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 (4-9) The above formula can be used for both low-pass and high-pass filters. Fig. 1-7: One-pole (first-order) low-pass filter.

The number of poles determines the roll-off rate of the filter
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) filter has a roll-off of -20 dB/decade; two-pole (second-order) filter has a roll-off of -40 dB/decade; three-pole (third-order) filter has a roll-off of -60 dB/decade; and so on.

Fig. 1-8: Three-pole (third-order) low-pass filter.
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.

ACTIVE LOW-PASS FILTERS
Advantages of active filters over passive filters (R, L, and C elements only): 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. 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. 3. Easy to adjust over a wide frequency range without altering the desired response.

Fig. 1-9: Single-pole active low-pass filter and response curve.
Single-Pole Filter 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.

The close-loop voltage gain is set by the values of R1 and R2, so that
(4-10)

Sallen-Key Low-Pass Filter
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. There are two low-pass RC circuits that provide a roll-off of -40 dB/decade above fc (assuming a Butterworth characteristics). One RC circuit consists of RA and CA, and the second circuit consists of RB and CB. Fig. 1-10: Basic Sallen-Key low-pass filter.

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

Example 3 Determine the critical frequency of the Sallen-Key low- pass filter in Figure below, and set the value of R1 for an appropriate Butterworth response. Answer: fc = 7.23 kHz, R1 = 586Ω

Fig. 1-11: Cascaded low-pass filter: third-order configuration.
Cascaded Low-Pass Filters 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.

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

Fig. 1-7: One-pole (first-order) low-pass filter.
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, R1 = 24 kΩ, and R2 = 2.2 kΩ Fig. 1-7: One-pole (first-order) low-pass filter.

ACTIVE HIGH-PASS FILTERS
Single-Pole Filter 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.

Sallen-Key High-Pass Filter
Components RA, CA, RB, and CB 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, R1 and R2. Fig. 1-14: Basic Sallen-Key high-pass filter.

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.

Fig. 1-7: One-pole (first-order) high-pass filter.
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, R1 = 36 kΩ, and R2 = 3.3 kΩ Fig. 1-7: One-pole (first-order) high-pass filter. 37

ACTIVE BAND-PASS FILTERS
Cascaded Low-Pass and High-Pass Filters 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.

Both filters provide the roll-off rates of –40 dB/decade, indicated in Fig. 5-17.
The critical frequency of the high-pass filter, fC1 must be lower than that of the low-pass filter, fC2 to make the center frequency overlaps. Fig. 1-17: The composite response curve of a high-pass filter and a low-pass filter.

The lower frequency, fc1 of the pass-band is calculated as follows:
(4-12) The upper frequency, fc2 of the pass-band is determined as follows: (4-13) The center frequency, fo of the pass-band is calculated as follows: (4-14)

Multiple-Feedback Band-Pass Filter
Multiple-feedback band-pass filter is another type of filter configuration. The feedback paths of the filter are through R2 and C1. R1 and C1 provide the low-pass filter, and R2 and C2 provide the high-pass filter. The center frequency is given as Fig. 1-18: Multiple-feedback band-pass filter. (5-15)

For C1 = C2 = C, the resistor values can be obtained using the following formulas:
(4-16) (4-17) (4-18)

Example 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

Fig. 1-19: State-variable filter.
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.

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

The Q is set by the feedback resistors R5 and R6 as follows:
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 R5 and R6 as follows: (4-19) 46

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

ACTIVE BAND-STOP FILTERS
Band-stop filters reject a specified band of frequencies and pass all others. The response are opposite to that of a band-pass filter. Band-stop filters are sometimes referred to as notch filters. Multiple-Feedback Band Stop Filter Fig. 1-22: Multiple-feedback band-stop filter. This filter is similar to the band-pass filter in Fig except that R3 has been moved and R4 has been added.

State-Variable Band-Stop Filter
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.

~ End of chapter 4 ~