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

2. 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

3. 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.

4. 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.

5. 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.

6. 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

7. 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

8. 1st Order Low-Pass Filter

9. 1st Order Low-Pass Filter

10. 1st Order Low-Pass Filter
When
And

11. 1st Order Low-Pass Filter
When
And
is known as 3-dB or cutoff frequency (rad/s)

12. 1st Order Low-Pass Filter

13. 2nd Order Low-Pass Filter
…(1)
…(2)
(1) & (2)

14. 2nd Order Low-Pass Filter
where

15. 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

16. 1st Order High-Pass Filter
Where;

17. 1st Order High-Pass Filter

18. 1st Order High-Pass Filter
When

19. 1st Order High-Pass Filter

20. 2nd Order High-Pass Filter
where

21. 2nd Order High-Pass Filter4

22. 1st & 2nd Order Filter
Low-Pass Filter (LPF)
High-Pass Filter

23. 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

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

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

26. 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

27. 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.

28. 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

29. 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

30. 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

31. Filter Response Characteristics
Roll-off depends on number of poles

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

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

34. Active High-Pass Filters
A Single-Pole Filter

35. 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;

36. Active High-Pass Filters
At node V1:

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

38. Active High-Pass Filters
where
= Damping Factor

39. Active High-Pass Filters

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

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

42. Active Low-Pass Filters
A Single-Pole Filter

43. 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

44. Active Low-Pass Filters
&

45. 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

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

47. Solution
From the table:
Hence:

48. 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;

49. Solution

50. Solution
A Bessel 2nd order LPF

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

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

53. Solution
Critical frequency
Butterworth response, R1/R2 = 0.586

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

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

56. 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

57. Solution

58. 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

59. 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

60. 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

61. Active Band-Pass Filters

62. 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:

63. 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.

64. 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).

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

66. 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.

67. 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

68. 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

69. 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.

70. 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.

71. 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.