Chapter 5: Active Filters EMT 359/3 – Analog Electronic II

Outlines Introduction Advantages of Active Filters over Passive Filters Types of filter Filter Response Characteristic Active Low-Pass Filter Active High-Pass Filter Active Band-Pass Filter Active Band-Stop Filter Summary

Introduction Filters are circuits that are capable of passing signals within a band of frequencies while rejecting or blocking signals of frequencies outside this band. This property of filters is also called “frequency selectivity”. Passive filters - built using components such as resistors, capacitors and inductors. Active filters - employ transistors or op-amps in addition to resistors and capacitors. Active filters are mainly used in communication and signal processing circuits. They are also employed in a wide range of applications such as entertainment, medical electronics, etc.

Advantages of Active Filters over Passive Filters Active filters can be designed to provide required gain, and hence no attenuation as in the case of passive filters. No loading problem, because of high input resistance and low output resistance of op-amp. Active Filters are cost effective as a wide variety of economical op-amps are available.

Active Filter Low-Pass filters High-Pass filters Band-Pass filters There are 4 basic categories of active filters: Low-Pass filters High-Pass filters Band-Pass filters Band-Stop filters Each of these filters can be built by using op-amp or transistor as the active element combined with RC, RL or RLC circuit as the passive elements.

Active Filter The passband is the range of frequencies that are allowed to pass through the filter. The critical frequency, fc is specified at the point where the response drops by 3 dB from the passband response (i.e. to 70.7% of the passband response) The stopband is the range of frequencies that have the most attenuation The transition region is the area where the fall-off occurs

Low-Pass Filter Ideal response Actual response Allows the frequency from 0 Hz to critical frequency, fH (also known as cutoff frequency) Ideally, the response drops abruptly at the critical frequency fH Ideal response Actual response

1st Order Low-Pass Filter

1st Order Low-Pass Filter

1st Order Low-Pass Filter When And

1st Order Low-Pass Filter When And is known as 3-dB or cutoff frequency (rad/s)

1st Order Low-Pass Filter

2nd Order Low-Pass Filter …(1) …(2) (1) & (2)

2nd Order Low-Pass Filter where

High-Pass Filter Ideal response Actual response Allows the frequencies above the critical frequency fL. (also known as the cutoff frequency. Ideally, the response rises abruptly at the critical frequency. Ideal response Actual response

1st Order High-Pass Filter Where;

1st Order High-Pass Filter

1st Order High-Pass Filter When

1st Order High-Pass Filter

2nd Order High-Pass Filter where

2nd Order High-Pass Filter 4

1st & 2nd Order Filter Low-Pass Filter (LPF) High-Pass Filter

Band-Pass Filter Ideal response Actual response Allows frequencies between a lower cutoff frequency (fL) and an upper cutoff frequency (fH). Ideal response Actual response

Band-Pass Filter Bandwidth (BW) Center frequency Quality factor (Q) is the ratio of center frequency fo to the BW

Band-Stop Filter Ideal response Actual response Frequencies below fc1 (fL) and above fc2 (fH) are passed. Ideal response Actual response

Filter Response Characteristics Identified by the shape of the response curve Passband flatness Attenuation of frequency outside the passband Three types: 1. Butterworth 2. Bessel 3. Chebyshev

Filter Response Characteristics Butterworth Amplitude response is very flat in passband. The roll-off rate -20 dB per decade (per filter order). normally used when all frequencies in the passband must have the same gain. Chebyshev overshoot or ripples in the passband. The roll-off rate greater than –20 dB. can be implemented with fewer poles and less complex circuitry for a given roll-off rate. Bessel Linear phase response. Ideal for filtering pulse waveforms.

Filter Response Characteristics Damping Factor determines the type of response characteristic either Butterworth, Chebyshev, or Bessel. The output signal is fed back into the filter circuit with negative feedback determined by the combination of R1 and R2

Filter Response Characteristics Critical Frequency 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: The above formula can be used for both low-pass and high-pass filters

Filter Response Characteristics Roll-off rate Greater roll-off rates can be achieved with more poles. Each RC set of filter components represents a pole. Cascading of filter circuits also increases the poles which results in a steeper roll-off. Each pole represents a –20 dB/decade increase in roll-off

Filter Response Characteristics Roll-off depends on number of poles

Active High-Pass Filters At critical frequency, Resistance = capacitive reactance critical frequency:

Active High-Pass Filters Roll-off depends on number of poles.

Active High-Pass Filters A Single-Pole Filter

Active High-Pass Filters The Sallen-Key second-order (two-pole) filter roll-off -40dB per decade Lets RA = RB = R and CA = CB = C;

Active High-Pass Filters At node V1:

Active High-Pass Filters ……(1) At node V2: ……(2) (2) in (1):

Active High-Pass Filters where = Damping Factor

Active High-Pass Filters

Active High-Pass Filters Cascaded HPF – Six pole cascade 3 Sallen-Key two-pole stages roll-off -120 dB per decade

Active Low-Pass Filters At critical frequency, Resistance = capacitive reactance critical frequency:

Active Low-Pass Filters A Single-Pole Filter

Active Low-Pass Filters The Sallen-Key second-order (two-pole) filter roll-off -40dB per decade For RA = RB = R and CA = CB = C

Active Low-Pass Filters &

Active Low-Pass Filters The 3-dB frequency c is related to o by a factor known as the FREQUENCY CORRECTION FACTOR (kLP), thus; Parameter table for Sellen-Key 2nd order LPF Type of response  kLP Bessel 1.732 0.785 Butterworth 1.414 1 Chebyshev (1 dB) 1.054 1.238 Chebyshev (2 dB) 0.886 1.333 Chebyshev (3 dB) 0.766 1.390

Example Design a Bessel 2nd order low-pass filter with a 3-dB frequency of 5 kHz. Use C = 22 nF.

Solution From the table: Hence:

Solution From the parameter table; In order to minimize the offset error, we have to meet the following condition: Or; And: From the parameter table;

Solution

Solution A Bessel 2nd order LPF

Frequency response curve of a 2nd order Bessel LPF Solution fc Frequency response curve of a 2nd order Bessel LPF

Example Determine critical frequency Set the value of R1 for Butterworth response

Solution Critical frequency Butterworth response, R1/R2 = 0.586

Active Low-Pass Filters Cascaded LPF – Three-pole cascade two-pole and single-pole roll-off -60dB per decade

Active Low-Pass Filters Cascaded LPF – Four pole cascade two-pole and two-pole roll-off -80dB per decade

Example Determine the capacitance values required to produce a critical frequency of 2680 Hz if all resistors in RC low pass circuit is 1.8 k

Solution

Active Low-Pass Filters SELLEN-KEY LPF The slope (roll-off) of a filter is associated with the its order – the higher the order of a filter, the steeper will be its slope. The slope increases by 20 dB for each order, thus; ORDER SLOPE (dB/decade) 1 20 2 40 3 60 4 80

Active Low-Pass Filters Active filters can be cascaded to increase its order, e.g. cascading a 2nd order and a 1st order filter will produce a 3rd order filter 2nd order LPF 1st order LPF Input signal Output signal Cascading a 2nd order and a 1st order filter produces a 3rd order filter with a roll-off of 60dB/decade

Active Band-Pass Filters A cascade of a low-pass and high-pass filter Band-pass filter formed by cascading a two-pole high-pass and a two- pole low-pass filters

Active Band-Pass Filters

Active Band-Pass Filters Multiple-Feedback BPF The low-pass circuit consists of R1 and C1. The high-pass circuit consists of R2 and C2. The feedback paths are through C1 and R2. Center frequency:

Active Band-Pass Filters Multiple-Feedback BPF For C1 = C2 = C, the resistor values can be obtained using the following formulas: The maximum gain, Ao occurs at the center frequency.

Active Band-Pass Filters State-Variable BPF It consists of a summing amplifier and two integrators. It has outputs for low-pass, high-pass, and band-pass. The center frequency is set by the integrator RC circuits. R5 and R6 set the Q (bandwidth).

Active Band-Pass Filters The band-pass output peaks sharply the center frequency giving it a high Q.

Active Band-Pass Filters 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.

Active Band-Stop Filters The BSF is opposite of BPF in that it blocks a specific band of frequencies. The multiple-feedback design is similar to a BPF with exception of the placement of R3 and the addition of R4

Active Band-Stop Filters 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

Summary The bandwidth of a low-pass filter is the same as the upper critical frequency. The bandwidth of a high-pass filter extends from the lower critical frequency up to the inherent limits of the circuit. The band-pass passes frequencies between the lower critical frequency and the upper critical frequency. A band-stop filter rejects frequencies within the upper critical frequency and upper critical frequency.

Summary The Butterworth filter response is very flat and has a roll-off rate of –20 dB. The Chebyshev filter response has ripples and overshoot in the passband but can have roll-off rates greater than –20 dB. The Bessel response exhibits a linear phase characteristic, and filters with the Bessel response are better for filtering pulse waveforms.

Summary A filter pole consists of one RC circuit. Each pole doubles the roll-off rate. The Q of a filter indicates a band-pass filter’s selectivity. The higher the Q the narrower the bandwidth. The damping factor determines the filter response characteristic.