Electronic Circuit DKT 214 Chapter 4 Active Filters By Pn. Hazila Othman By.

Electronic CircuitElectronic CircuitDKT 214DKT 214

Chapter 4Chapter 4

Active FiltersActive Filters

ByBy

Pn. Hazila OthmanPn. Hazila OthmanByBy

Pn. Hazila OthmanPn. Hazila Othman

FilterFilter is a circuit used for signal processing due to its capability of passingpassing signals with certain selected frequencies and rejectingrejecting or attenuatingattenuating signals with other frequencies. This property is called selectivityselectivity.

Filter can be passive or active filter.

Passive filtersPassive filters: The circuits built using RC, RL, or RLC circuits.

Active filtersActive filters : The circuits that employ one or more op- amps in the design an addition to resistors and capacitors.

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.

Fig. 1-1: Low-pass filter responses

Vo

A low-pass filterlow-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.

PassbandPassband 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 regionTransition region shows the area where the fall-off occurs.

StopbandStopband is the range of frequencies that have the most attenuation.

Critical frequencyCritical frequency, ffcc, (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.

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.

VoVin

The bandwidthbandwidth of an idealideal low-pass filter is equal to ffcc:

cfBW

When XXCC = R = R, the critical frequency of a low-pass RC filter can be calculated using the formula below:

RCfc 2

1

(4-1)

(4-2)

A high-pass filterhigh-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 XXCC = R = R, where

RCfc 2

1 (4-3)

A band-pass filterband-pass filter passes all signals lying within a band between a lower-lower-frequency limitfrequency limit and upper-frequency limitupper-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)bandwidth (BW) is defined as the differencedifference between the upper critical upper critical frequency (ffrequency (fc2c2)) and the lower lower critical frequency (fcritical frequency (fc1c1)).

12 cc ffBW

21 cco fff

The frequency about which the passband is centered is called the center center frequencyfrequency, ffoo, defined as the geometric mean of the critical frequencies.

(4-4)

(4-5)

The quality factor (Q)quality factor (Q) of a band-pass filter is the ratio of the center frequency to the bandwidth.

BW

fQ o

The quality factor (Q) can also be expressed in terms of the damping factor (DF) of the filter as

DFQ

1

(4-6)

(4-7)

Band-stop filterBand-stop filter is a filter which its operation is oppositeopposite to that of the band-pass filter because the frequencies withinwithin the bandwidth are rejectedrejected, and the frequencies outsideoutside bandwidth are passedpassed. Its also known as notch, band-reject or band-elimination filter

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

The characteristics of filter response can be ButterworthButterworth, ChebyshevChebyshev, or BesselBessel characteristic.

Fig. 1-5: Comparative plots of three types of filter response characteristics.

Butterworth characteristicButterworth characteristic

Filter response is characterized by flat amplitude responseflat 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 gainsame gain.

Chebyshev characteristicChebyshev characteristic

Filter response is characterized by overshootovershoot or ripplesripples in the passband. Provides a roll-off rate greater than -20 dB/decade/pole.

Filters with the Chebyshev response can be implemented with fewer polesfewer poles and less complex less complex circuitrycircuitry for a given roll-off rate.

Bessel characteristicBessel characteristic

Filter response is characterized by a linear characteristiclinear 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 characteristicBessel characteristic

Filter response is characterized by a linear characteristiclinear 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.

The damping factor (DF)damping factor (DF) primarily determines if the filter will have a Butterworth, Chebyshev, or Bessel response.

This active filter consists of an an amplifieramplifier, a negative feedback a negative feedback circuitcircuit and RC circuitRC circuit.

The amplifier and feedback are connected in a non-inverting non-inverting configurationconfiguration.

DF is determined by the negative feedback and defined as

2

12R

RDF Fig. 1-6: Diagram of an active filter.

(4-8)

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 1.586 (4.0dB).

The critical frequencycritical frequency, ffcc 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

RCfc 2

1

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)

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.

The number of filter poles can be increased by cascadingcascading. 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.

Advantages of active filters over passive filters (R, L, and C elements only):

1. By containing the op-amp, active filters can be designed to provide required gain, and hence no signal attenuationno signal attenuation as the signal passes through the filter.

2. No loading problem 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 rangeEasy to adjust over a wide frequency range without altering the desired response.

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

12

1)( R

RA NIcl

(4-10)

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 fc (assuming a Butterworth characteristics).

One RC circuit consists of RA and CA, and the second circuit consists of RB and CB.

The critical frequency for the Sallen-Key filter is

BABA

cCCRR

f2

1 (4-11)

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.

Four-pole filter is obtained by cascading Sallen-Key (2-pole) filters.

Fig. 1-12: Cascaded low-pass filter: fourth-order configuration.

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.

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.

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:

1111

12

1

BABA

CCCRR

f

2222

22

1

BABA

CCCRR

f

The upper frequency, fc2 of the pass-band is determined as follows:

The center frequency, fo of the pass-band is calculated as follows:

21 CCo fff

(4-12)

(4-13)

(4-14)

Multiple-feedback band-pass filter is another type of filter configuration.

The feedback paths of the filter are through R1 and C1.

RR11 and CC11 provide the low-low-pass filterpass filter, and RR22 and CC22 provide the high-pass high-pass filterfilter.

The center frequency is given as

21231 )//(2

1

CCRRRfo

Fig. 1-18: Multiple-feedback band-pass filter.

(5-15)

For C1 = C2 = C, the resistor values can be obtained using the following formulas:

ooCAf

QR

21

Cf

QR

o2

)2(2 23oo AQCf

QR

The maximum gain, Ao occurs at the center frequency.

(4-16)

(4-17)

(4-18)

State-variable filterState-variable filter contains a summing amplifiera summing amplifier and two op-amp integratorstwo 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.

In the state-variable filter, the bandwidthbandwidth is dependentdependent on the critical frequency and the quality quality factorfactor, QQ is independentindependent on the critical frequency.

The Q is set by the feedback resistors R5 and R6 as follows:

1

3

1

6

5

R

RQ (4-19)

Biquad filterBiquad filter contains an integratoran integrator, followed by an an inverting amplifierinverting amplifier, and then an integratoran integrator.

In a biquad filter, the bandwidthbandwidth is independentindependent and the QQ is dependentdependent on the critical frequency.

Fig. 1-21: A biquad filter.

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 notch filtersfilters.

This filter is similar to the band-pass filter in Fig. 1-18 except that RR33 has has been moved and Rbeen moved and R44 has has been addedbeen added.

Fig. 1-22: Multiple-feedback 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.

Example-1:Example-1:

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 = 0.022 μF, R = 3.3 kΩ, R1 = 24 kΩ, and R2 = 2.2 kΩ

Fig. 1-7: One-pole (first-order) low-pass filter.

Example-2:Example-2:

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.

Example-3:Example-3:Determine the critical frequency, the pass-band gain, and damping factor for the second-order low-pass active filter with the following circuit components:

R1 = R2 = 3 kΩ, C1 = C2 = 0.05 μF, R1 = 10 kΩ, R2 = 15 kΩ

Fig. 1-10: Basic Sallen-Key/second-order low-pass filter.

Example-4:Example-4:

Determine the capacitance value required to produce a critical frequency of 2.68k Hz if all the resistors in the RC low-pass circuits are 1.8kΩ. Also select values for the feedback resistors to get a Butterworth response.