(EC1406)

ACTIVE FILTERSBy: Dr. Mayank Srivastava, Dept. of

ECE, NIT JSR

B S MUNDA

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.

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.

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 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.

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.

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 bandwidth of an ideal low-pass filter is equal to fc:

cfBW

RCfc

2

1

When XC = R, the critical frequency of a low-pass RC filter

can be calculated using the formula below:

(4-1)

(4-2)

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

RCfc

2

1 (4-3)

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).

12 cc ffBW

21 cco fff

The frequency about which the passband is centered is called the

center frequency, fo, defined as the geometric mean of the

critical frequencies.

(4-4)

(4-5)

The quality factor (Q) of a band-pass filter is the ratio of

the center frequency to the bandwidth.

BW

fQ o

DFQ

1

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

(4-6)

(4-7)

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

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 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.

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.

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

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).

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

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

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) 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.

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.

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

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.

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)

If RA = RB = R and CA = CB = C, the critical frequency can be expressed as:

RCfc

2

1

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

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.

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 = 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.

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.

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.

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 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

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 23

oo AQCf

QR

(4-16)

(4-17)

(4-18)

1

2

2R

RAo

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

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.

In the state-variable filter, the bandwidth is dependent on the critical frequency and the quality factor, Q is independenton 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 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.

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.

This filter is similar to the band-pass filter in Fig. 1-18 except that R3 has been moved and R4 has been 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.