Electronic DevicesNinth Edition

Floyd

Electronic DevicesNinth Edition

Floyd

Electronic DevicesNinth Edition

Floyd

Chapter 15

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 =

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

calculated using the formula below:

RCfc π2

1=

(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 π2

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=

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 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.

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.

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).

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 π2

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

fπ2

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

fπ

=

2222

22

1

BABA

CCCRR

fπ

=

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.

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 π

=

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

π21 =

Cf

QR

oπ=2

)2(2 23oo AQCf

QR

−=

π

The maximum gain, Ao occurs at the center frequency.

(4-16)

(4-17)

(4-18)

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.

Filter Measurements

Filter responses can be observed in practical circuits with a swept frequency measurement. The test setup for this measurement is shown here.

X

Sweepgenerator Vin

Filter

Oscilloscope

Y

Sawtoothoutput

Vout

The sawtooth waveform synchronizes the oscilloscope with the sweep generator.

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:

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:

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:

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.