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Active Filters
IntroductionIntroduction
 Filters are circuits that are capable of passing signals
within a band of frequencies while rej...
Types of Filters
There are two broad categories of filters:
 An analog filter processes continuous-time signals
 A digi...
Ideal Filters
Passband Stopband
Stopband Passband
Passband PassbandStopband
Lowpass Filter Highpass Filter
Bandstop Filter...
Analog Filter Responses
H(f)
ffc
0
H(f)
ffc
0
Ideal “brick wall” filter Practical filter
Filter can be also be categorized as passive or active..
Passive filtersPassive filters: The circuits built using RC, RL,...
Passive filters
Passive filters use resistors, capacitors, and inductors
(RLC networks).
To minimize distortion in the f...
Active filters overcome these drawbacks and are
realized using resistors, capacitors, and active
devices (usually op-amps...
Advantages
Advantages of active RC filters include:
 reduced size and weight
 increased reliability and improved perfor...
Disadvantages
Active RC filters also have some disadvantages:
 limited bandwidth of active devices limits the highest
at...
Bode Plots
Bode plots are important when considering the
frequency response characteristics of amplifiers.
 They plot th...
Bode plots use a logarithmic scale for
frequency.
where a decade is defined as a range of
frequencies where the highest an...
The decibel (dB)
Two levels of power can be compared using a
unit of measure called the bel.
The decibel is defined as:
1 ...
A common dB term is the half power point
which is the dB value when the P2 is one-
half P1.
1
2
10log10
P
P
dB =
dBdB 301....
15
Decibel (dB) By Definition: 





=
1
2
10log10
P
P
dB
(1) Power Gain in dB :






=
in
o
p
P
P
dBA 10...
16
Cascaded System
Av1 Av2 Av3
x10 x10x10
vin
vout
20dB 20dB 20dB
321 vvvv
AAAA ××=
3
10101010 =××=v
A
( )32110
log20)( vv...
Poles & Zeros of the transfer
functionpole—value of s where the denominator goes to
zero.
zero—value of s where the nume...
Actual response
Vo
 A low-pass filterlow-pass filter is a filter that passes frequencies from 0Hz to
critical frequency, ...
StopbandStopband is the range of frequencies that have the most attenuation.
Critical frequencyCritical frequency, ffcc, (...
 At low frequencies, XC is very high and the capacitor circuit can be
considered as open circuit. Under this condition, V...
 The bandwidthbandwidth of an idealideal low-pass filter is equal to ffcc:
cfBW =
The critical frequency of a low-pass R...
 A high-pass filterhigh-pass filter is a filter that significantly attenuates or rejects
all frequencies below fc and pas...
 The critical frequency of a high-pass RC filter occurs when
XXCC = R= R and can be calculated using the formula below:
R...
 A band-pass filterband-pass filter passes all signals lying within a band
between a lower-frequency limitlower-frequency...
 The bandwidth (BW)bandwidth (BW) is defined as the differencedifference between
the upper critical frequency (fupper cri...
21 cco fff =
 The frequency about which the pass band is centered is called
the center frequencycenter frequency, ffoo an...
 Band-stop filterBand-stop filter is a filter which
its operation is oppositeopposite to that of
the band-pass filter bec...
RC
fc
π2
1
=
cXR =
 Figure below shows the basic Low-Pass filter circuit
Cf
R
cπ2
1
=
C
R
cω
1
=
At critical frequency,
R...
RC
fc
π2
1
=
cXR =
 Figure below shows the basic High-Pass filter circuit :
Cf
R
cπ2
1
=
C
R
cω
1
=
At critical frequency...
Single-Pole Passive Filter
First order low pass filter
Cut-off frequency = 1/RC rad/s
Problem : Any load (or source) im...
Ref:080222HKN EE3110 Active Filter (Part 1)31
Bode Plot (single pole)






+
=
+
=
o
j
CRj
jH
ω
ωω
ω
1
1
1
1
)(
2...
32
dB
jH )( ω
(log)ω
x
ω x
ω2 x
ω10
6dB20dB
slope
-6dB/octave
-20dB/decade






−≈
o
jH
ω
ω
ω 10log20)(
For octav...
Single-Pole Active Low-Pass Filter
Same frequency response as passive filter.
Buffer amplifier does not load RC network....
Single-pole active low-pass filter and response curve.
 This filter provides a roll-off rate of -20 dB/decade above the
c...
The op-amp in single-pole filter is connected as a
noninverting amplifier with the closed-loop voltage gain in the
passba...
 The critical frequencycritical frequency, ffcc is determined by the values of R
and C in the frequency-selective RC circ...
 In high-pass filters, the roles of the capacitorcapacitor and resistorresistor are
reversedreversed in the RC circuits a...
The op-amp in single-pole filter is connected as a
noninverting amplifier with the closed-loop voltage gain in the
passba...
The number of poles determines the roll-off rate of the filter.
 A Butterworth response produces -20dB/decade/pole
This ...
 The number of filter poles can be increased by cascadingcascading.
To obtain a filter with three poles, cascade a two-po...
41
Two-Stage Band-Pass Filter
R2 R1
vin
C1
C2
Rf1
Rf2
C4 C3
R3
R4
+V
-V
vout
Rf3
Rf4
+
-
+
-
+V
-V
Stage 1
Two-pole low-pa...
42
Band-Stop (Notch) Filter
The notch filter is designed to block all frequencies that fall
within its bandwidth. The circ...
43
Notch filter
44
Transfer function H(jω)
Transfer
Function
)( ωjH
VoVi
)(
)(
)(
ω
ω
ω
jV
jV
jH
i
o=
)Im()Re( HjHH +=
22
)Im()Re( HHH +=
45
Frequency transfer function of filter
H(jω)
HL
HL
o
o
o
o
ffffjH
fffjH
ffjH
ffjH
ffjH
ffjH
><=
<<=
>=
<=
>=
<=
and0)(
1...
Advantages of active filters over passive filters (R, L, and C
elements only):
1. By containing the op-amp, active filters...
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Active filters

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Active filters

  1. 1. Active Filters
  2. 2. IntroductionIntroduction  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”.
  3. 3. Types of Filters There are two broad categories of filters:  An analog filter processes continuous-time signals  A digital filter processes discrete-time signals. The analog or digital filters can be subdivided into four categories:  Lowpass Filters  Highpass Filters  Bandstop Filters  Bandpass Filters
  4. 4. Ideal Filters Passband Stopband Stopband Passband Passband PassbandStopband Lowpass Filter Highpass Filter Bandstop Filter PassbandStopband Stopband Bandpass Filter M(ω) M(ω) ω ω ω ω ω c ω c ω c 1 ω c 1 ω c 2 ω c 2
  5. 5. Analog Filter Responses H(f) ffc 0 H(f) ffc 0 Ideal “brick wall” filter Practical filter
  6. 6. Filter can be also be categorized as passive or active.. 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
  7. 7. Passive filters Passive filters use resistors, capacitors, and inductors (RLC networks). To minimize distortion in the filter characteristic, it is desirable to use inductors with high quality factors practical inductors includes a series resistance.  They are particularly non-ideal  They are bulky and expensive
  8. 8. Active filters overcome these drawbacks and are realized using resistors, capacitors, and active devices (usually op-amps) which can all be integrated:  Active filters replace inductors using op-amp based equivalent circuits.
  9. 9. Advantages Advantages of active RC filters include:  reduced size and weight  increased reliability and improved performance  simpler design than for passive filters and can realize a wider range of functions as well as providing voltage gain  in large quantities, the cost of an IC is less than its passive counterpart
  10. 10. Disadvantages Active RC filters also have some disadvantages:  limited bandwidth of active devices limits the highest attainable frequency (passive RLC filters can be used up to 500 MHz)  require power supplies (unlike passive filters)  increased sensitivity to variations in circuit parameters caused by environmental changes compared to passive filters For many applications, particularly in voice and data communications, the economic and performance advantages of active RC filters far outweigh their disadvantages.
  11. 11. Bode Plots Bode plots are important when considering the frequency response characteristics of amplifiers.  They plot the magnitude or phase of a transfer function in dB versus frequency.
  12. 12. Bode plots use a logarithmic scale for frequency. where a decade is defined as a range of frequencies where the highest and lowest frequencies differ by a factor of 10. 10 20 30 40 50 60 70 80 90 100 200 One decade
  13. 13. The decibel (dB) Two levels of power can be compared using a unit of measure called the bel. The decibel is defined as: 1 bel = 10 decibels (dB) 1 2 10log P P B =
  14. 14. A common dB term is the half power point which is the dB value when the P2 is one- half P1. 1 2 10log10 P P dB = dBdB 301.3 2 1 log10 10 −≈−=
  15. 15. 15 Decibel (dB) By Definition:       = 1 2 10log10 P P dB (1) Power Gain in dB :       = in o p P P dBA 10 log10)(       = in in P P dB 10 log100           =− in in P P dB 2 1 log103 10       =+ in in P P dB 2 log103 10 Pin Pout (2) Voltage Gain in dB: (P=V2 /R) vin vout       = in o v v v dBA 10 log20)(       = in in v v dB 10 log200           =− in in v v dB 2 1 log206 10       =+ in in v v dB 2 log206 10
  16. 16. 16 Cascaded System Av1 Av2 Av3 x10 x10x10 vin vout 20dB 20dB 20dB 321 vvvv AAAA ××= 3 10101010 =××=v A ( )32110 log20)( vvvv AAAdBA ××= ( ) ( ) ( )310210110 log20log20log20)( vvvv AAAdBA ++= ( ) ( ) ( )dBAdBAdBAdBA vvvv 321 )( ++= dBdBdBdBAv 202020)( ++= dBdBAv 60)( = ( ) dB2010log20 10 = ( ) dB6010log20 3 10 =
  17. 17. Poles & Zeros of the transfer functionpole—value of s where the denominator goes to zero. zero—value of s where the numerator goes to zero.
  18. 18. Actual response Vo  A low-pass filterlow-pass filter is a filter that passes frequencies from 0Hz to critical frequency, fc and significantly attenuates all other frequencies. Ideal response  Ideally, the response drops abruptly at the critical frequency, fH roll-off rateroll-off rate
  19. 19. 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. 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. roll-off rateroll-off rate
  20. 20.  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 gainfalls and drops off gradually as the frequency is increased. Vo
  21. 21.  The bandwidthbandwidth of an idealideal low-pass filter is equal to ffcc: cfBW = The critical frequency of a low-pass RC filter occurs when XXCC = R= R and can be calculated using the formula below: RC fc π2 1 =
  22. 22.  A high-pass filterhigh-pass filter is a filter that significantly attenuates or rejects all frequencies below fc and passes all frequencies above fc.  The passband of a high-pass filter is all frequencies above the critical frequency.. Vo Actual response Ideal response  Ideally, the response rises abruptly at the critical frequency, fL
  23. 23.  The critical frequency of a high-pass RC filter occurs when XXCC = R= R and can be calculated using the formula below: RC fc π2 1 =
  24. 24.  A band-pass filterband-pass filter passes all signals lying within a band between a lower-frequency limitlower-frequency limit and upper-frequency limitupper-frequency limit and essentially rejects all other frequencies that are outside this specified band. Actual response Ideal response
  25. 25.  The bandwidth (BW)bandwidth (BW) is defined as the differencedifference between the upper critical frequency (fupper critical frequency (fc2c2)) and the lower criticallower critical frequency (ffrequency (fc1c1)). 12 cc ffBW −=
  26. 26. 21 cco fff =  The frequency about which the pass band is centered is called the center frequencycenter frequency, ffoo and defined as the geometric mean of the critical frequencies.
  27. 27.  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 above ffc1c1 and ffc2c2 are passedpassed. Actual response For the band-stop filter, the bandwidthbandwidth is a band of frequencies between the 3 dB points, just as in the case of the band-pass filter response. Ideal response
  28. 28. RC fc π2 1 = cXR =  Figure below shows the basic Low-Pass filter circuit Cf R cπ2 1 = C R cω 1 = At critical frequency, Resistance = Capacitance So, critical frequency ;
  29. 29. RC fc π2 1 = cXR =  Figure below shows the basic High-Pass filter circuit : Cf R cπ2 1 = C R cω 1 = At critical frequency, Resistance = Capacitance So, critical frequency ;
  30. 30. Single-Pole Passive Filter First order low pass filter Cut-off frequency = 1/RC rad/s Problem : Any load (or source) impedance will change frequency response. vin vout C R RCs RC sCR sCR sC ZR Z v v C C in out /1 /1 1 1 /1 /1 + = + = + = + =
  31. 31. Ref:080222HKN EE3110 Active Filter (Part 1)31 Bode Plot (single pole)       + = + = o j CRj jH ω ωω ω 1 1 1 1 )( 2 1 1 )(       + = o jH ω ω ω               +== 2 1010 11log20)(log20)( o dB jHjH ω ω ωω ⇒       −≈ o dB jH ω ω ω 10 log20)( For ω>>ωo R C VoVi Single pole low-pass filter
  32. 32. 32 dB jH )( ω (log)ω x ω x ω2 x ω10 6dB20dB slope -6dB/octave -20dB/decade       −≈ o jH ω ω ω 10log20)( For octave apart, 1 2 = o ω ω dBjH 6)( −≈ω For decade apart, 1 10 = oω ω dBjH 20)( −≈ω
  33. 33. Single-Pole Active Low-Pass Filter Same frequency response as passive filter. Buffer amplifier does not load RC network. Output impedance is now zero. vin vout C R
  34. 34. Single-pole active low-pass filter and response curve.  This filter provides a roll-off rate of -20 dB/decade above the critical frequency.
  35. 35. The op-amp in single-pole filter is connected as a noninverting amplifier with the closed-loop voltage gain in the passband is set by the values of R1 and R2 : 1 2 1 )( += R R A NIcl  The critical frequency of the single-pole filter is : RC fc π2 1 =
  36. 36.  The critical frequencycritical frequency, ffcc is determined by the values of R and C in the frequency-selective RC circuit.  Each RCRC set of filter components represents a polepole.  Greater roll-off ratesGreater roll-off rates can be achieved with more polesmore poles.  Each pole represents a -20dB/decade-20dB/decade increase in roll-off. One-pole (first-order) low-pass filter.
  37. 37.  In high-pass filters, the roles of the capacitorcapacitor and resistorresistor are reversedreversed in the RC circuits as shown from Figure (a). The negative feedback circuit is the same as for the low-pass filters.  Figure (b) shows a high-pass active filter with a -20dB/decade roll-off Single-pole active high-pass filter and response curve.
  38. 38. The op-amp in single-pole filter is connected as a noninverting amplifier with the closed-loop voltage gain in the passband is set by the values of R1 and R2 : 1 2 1 )( += R R A NIcl The critical frequency of the single-pole filter is : RC fc π2 1 =
  39. 39. The number of poles determines the roll-off rate of the filter.  A Butterworth response produces -20dB/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
  40. 40.  The number of filter poles can be increased by cascadingcascading. To obtain a filter with three poles, cascade a two-pole with one-pole filters. Three-pole (third-order) low/high pass filter.
  41. 41. 41 Two-Stage Band-Pass Filter R2 R1 vin C1 C2 Rf1 Rf2 C4 C3 R3 R4 +V -V vout Rf3 Rf4 + - + - +V -V Stage 1 Two-pole low-pass Stage 2 Two-pole high-pass BW f1 f2 f Av Stage 2 response Stage 1 response fo BW = f2 – f1 Q = f0 / BW
  42. 42. 42 Band-Stop (Notch) Filter The notch filter is designed to block all frequencies that fall within its bandwidth. The circuit is made up of a high pass filter, a low-pass filter and a summing amplifier. The summing amplifier will have an output that is equal to the sum of the filter output voltages. f 1 f 2 v in v out Lowpass filter Highpass filter Summing amplifier Σ -3dB{ f f2 f1 Av(dB) low-pass high-pass Block diagram Frequency response
  43. 43. 43 Notch filter
  44. 44. 44 Transfer function H(jω) Transfer Function )( ωjH VoVi )( )( )( ω ω ω jV jV jH i o= )Im()Re( HjHH += 22 )Im()Re( HHH +=
  45. 45. 45 Frequency transfer function of filter H(jω) HL HL o o o o ffffjH fffjH ffjH ffjH ffjH ffjH ><= <<= >= <= >= <= and0)( 1)( FilterPass-Band(III) 1)( 0)( FilterPass-High(II) 0)( 1)( FilterPass-Low(I) ω ω ω ω ω ω responsephasespecificahas allfor1)( Filtershift)-phase(orPass-All(V) and1)( 0)( Filter(Notch)Stop-Band(IV) fjH ffffjH fffjH HL HL = ><= <<= ω ω ω
  46. 46. 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 problemNo 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.

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