Chapter 6:

Active Filters

Active Filters

1. Basic filter responses 2. Filter response characteristic-Butterworth,

Chebychev, Bessel 3. Active low pass filter 4. Active high pass filter 5. Active band pass filter 6. Active band stop filter 7. Filter response measurement

Active filters

Passing signals with certain selected frequencies while rejecting signals with other frequencies.

Provides controllable cutoff frequencies and controllable gain.

4 basic categories : low-pass, high-pass, band-pass and band-stop.

Basic Filter ResponsePassband – range of frequencies allowed to pass

through with minimum attenuation ( < -3 dB)Cutoff frequency – the end of the passband (point

where response drops -3 dB)Stopband – the end of the response.

Basic Filter Response

How close a tuned amplifier comes to having the characteristics of an ideal circuit depends on the quality (Q) of the circuit.

The Q of a tuned amplifier is a figure of merit that equals the ratio of its geometric center frequency to

its bandwidth. By formula:

Basic Filter Response

Active filter frequency-response curves

Active filter frequency-response

Each RC circuit is referred to as a pole. Thus, a one-pole filter contains one RC circuit, a two-pole filter contains two RC circuits, and so on.

The order of an active filter indicates the number of poles it contains.

The number of poles in a filter determines its ultimate roll-off rate.

Low-Pass Filter

Actual filter responses depend on the number of poles (number of RC circuits contained in the filter).

Roll-off rate with steeper transition region is good for better filtering of unwanted frequencies.

Exact response depends on type of filter and number of poles.

Low-Pass Filter

Basic LPF circuitRC

fc 21

High-Pass Filter

A high-pass filter effectively blocks below fc and allowing only the frequencies above fc to pass.

The same formula is used for the critical frequency for both low and high pass filters.

Band-Pass Filter

Allows frequencies between a lower critical frequency (fc1) and an upper critical frequency (fc2) to pass while effectively blocking all others.

BW = 12 cc ff

Centre frequency:

21 cco fff

Quality factor (Q) of band-pass filter. Higher Q means narrower bandwidth and better selectivity.

Narrow-band (Q >10) and wide-band (Q <10).

orBW

fQ o

DFQ

1

Band-stop FilterOpposite of a band-pass. Frequencies above and

below fc1 and fc2 are passed while effectively blocking the frequencies between.

Also known as band-reject, band-elimination filter.

Filter response characteristics

Identified by shape of

response curve.

Each type of filter response can be modified through circuit components values to have either Butterworth, Chebyshev or Bessel characteristics.

Butterworth

Very flat amplitude response in the passband and roll-off rate -20 dB/decade.

Phase response is non-linear and phase shift (time delay) varies nonlinearly with frequency.

Normally used when all frequencies in passband must have the same gain.

Chebyshev

Very rapid roll-off rate (greater than -20 dB/decade).

Filters can be designed with fewer poles and less complex circuitry.

Overshoot or ripples can be seen in the passband of the frequency response.

Bessel

Very linear phase characteristic (the phase shift increases linearly with frequency).

Almost no overshoot on the output and normally used for filtering pulse waveforms without distorting the shape of the input waveform.

Damping Factor

Damping Factor (DF) of an active filter circuit determines which response characteristics the filter exhibits (Butterworth, Chebyshev or Bessel).

The filter can be a low-pass, high-pass, band-pass or a band-stop type.

Filter Response Characteristics

The output signal is fed back into the filter circuit with negative feedback determined by the combination of R1 and R2. The negative feedback ultimately determines the type of filter response is produced. The equation below defines the damping factor.

DF = 2 - R1/R2

Cut-off frequency and Roll-off rate

Single pole (first-order) are the same for low and high-pass filters. The numbers of poles determine the roll-off rate. - One-pole has -20 dB roll-off - Two-pole has -40 dB - Three-pole has -60 dB and so on

To obtain a filter with three poles (third-order) or more, one-pole or two-pole filters are cascaded.Example:

i)Third order filter → cascade a second order and a first order filters ii) Fourth order filter → cascade two second order filters.

Cut-off frequency and Roll-off rate

Filter Response Characteristics

For 3-pole filter, cascade 2-pole LP and 1-pole LP (-60 dB/dec)

For 4-pole filter, cascade 2-pole LP and 2-pole LP (-80 dB/dec)

Butterworth characteristic is the most widely used because of its maximum flat response.

Table above shows roll-off rates, damping factors, and feedback resistor ratios for up to sixth-order Butterworth filters.

ExampleDetermine the capacitance values required to produce an fc=2680 Hz if all Rs are 1.8 kOHM. Select R feedback for Butterworth response.

Both stages must have the same fc = 2680 Hz.

Assuming equal-value capacitors;

fc = ½ RC Therefore, C = ½ Rf = 0.033 uF

CA1=CA2=CB1=CB2=0.033 uF

For Butterworth response,

Refer Table: (1st stage) DF=1.848 and R1/R2=0.152

Assuming R2=R4=1.8kΩ,

R1=0.152R2=274

Refer Table: (2nd stage) DF=0.765 and R3/R4=1.235

Assuming R1=270

R3=1.235R4=2.22 k

Example - solution

Active Low-pass filters – One pole

Non-inverting amplifier with closed-loop voltage gain in the pass band set by the values or R1 and R2.

12

1)( R

RA NIcl

Sallen-Key Low-pass filter – Two pole

BABA

cCCRR

f2

1

Cut-off frequency

If RA = RB = R and

CA = CB = C, then

RCfc 2

1

Cascaded Low-pass filter – Three & Four-pole

Active High-pass filter – One pole

The negative feedback circuit is as same as the low-pass filter with -20 dB/decade roll-off.

Sallen-Key High-pass filter –Two pole

The positions of the resistors and capacitors are opposite to those in low-pass configuration.

Cascading High-pass filter

Six-pole high-pass filter consisting of 3 Sallen-Key Two-pole stages (can achieved -120 dB/decade roll-off).

Active Band-Pass filter-Cascaded Low-Pass & High-Pass Filter

Band-pass filter formed by cascading a two-pole high-pass and a two-pole low-pass filter (does not matter in which order the filters are cascaded).

Cut-off Frequency

1111

12

1

BABA

cCCRR

f

2222

22

1

BABA

cCCRR

f

21 CCo fff

High-pass filter

Low-pass filter

Centre frequency

Multiple- feedback Band-pass filter

Maximum gain (Ao) at centre frequency

321

31

2

1

RRR

RR

Cfo

1

2

2R

RAO and

Active Band-stop filter

Multiple-feedback band-stop

State-variable band-stop

Problem 1

• Calculate the bandwidth of the filter

Problem 2

• Perform the complete analysis of the amplifier

The End………