1. Basic Filter Theory Review
Loading tends to make filter’s response very droopy, which is quite undesirable
To prevent such loading, filter sections may be isolated using high-input-impedance buffers
‘A’ is closed-loop gain of op amp
H(jf) dB = 20 log [A/{sqrt(1+(f/fc)2}] <-tan-1 (f/fC)
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

2. Higher-Order LP Filters
Higher-order filters may be realized by cascading basic RC sections
Higher the order of filter, more closely its response resembles that of ideal brick wall filter
Frequency response curves for LP filters
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

3. Second-Order LP Filters
Second-order LP filter may be designed using two cascaded RC sections or by using an LC section
LC filter is not restricted to one single response shape
Corner frequency of LC filter is given by
fc = 1/[2πSqrt(LC)]
A given LC product can be achieved using infinitely many different inductor and capacitor combinations giving much more flexibility in terms of response shape
Using high C to L ratio results in low damping coefficient (α) and peaking in response curve
Using low C to L ratio results in higher α and smoother response curve
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

4. Second-Order LP LC Filters
α = 1.414: response is as flat as possible in passband and is called critical damping
Lower α result in peaking near corner and more rapid attenuation in transition region ultimate rolloff is -40 dB/decade for second-order filter
Filters with flat response in passband: Butterworth filters
Filters with peaked response in passband: Chebyshev filters
Filters with α < 1.414: underdamped
Filters with α > 1.414: overdamped
Filters with α = 1.414: critically damped
α also affects location of fc
Critically damped filters: no effect
Underdamped filters: increase in fc
Overdamped filters: decrease in fc
Effect of Damping Coefficient on second-order LP LC filter
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

5. Second-Order LP LC Filters
Frequency scaling factors Kf indicated relative increase or decrease from fc of an equivalent filter with α = 1.414
fc of a second-order Butterworth active filter fc = 1/[2πSqrt(C1C2R1R2)]
If α is changed, new fc would be given by fc = Kf/[2πSqrt(C1C2R1R2)]
Bessel filter: provides nearly linear phase shift as function of frequency; has droopy passband response with gradual rolloff and very low overshoot for transient inputs (no ringing)
α = 1.732, Kf = 0.785
Butterworth filter: allowing flattest possible passband; most popular filter
α = 1.414, Kf = 1
Chebyshev filters: allow peaking in passband, with more rapid transition-region attenuation; higher the peaking, more nonlinear the phase response becomes, and more rapid the transition-region attenuation becomes; these filters tend to overshoot and ring in response to transients
1-dB Chebyshev filters α = 1.045, Kf = 1.159
2-dB Chebyshev filters α = 0.895, Kf = 1.174
3-dB Chebyshev filters α = 0.767, Kf = 1.189

6. Second-Order LP LC Filters
Effects of damping on phase response of second-order LP LC filters
Second-order filters are very frequently encountered in many applications
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

7. Second-Order LP HP Filters
Decibel gain magnitudes for second-order LP filters in terms of damping coefficient
H(jf) dB = 20 log [A/{Sqrt(1+(α2-2)(f/fc)2+(f/fc)4}]
For nth order Butterworth response (α = 1.414)
H(jf) dB = 20 log [A/{Sqrt(1+(f/fc)2n}]
Second-order HP filters
H(jf) dB = 20 log [A/{Sqrt(1+(α2-2)(fc/f)2+(fc/f)4}]
H(jf) dB = 20 log [A/{Sqrt(1+(fc/f)2n}]

8. Active LP and HP Filters
It is not possible to produce passive RC filter with α = 1.414
Using passive filter techniques, one must resort to inductor-capacitor designs in such cases
At low frequencies, inductors required to produce many response shapes tend to be excessively large, heavy, and expensive
Inductors generally tend to pick up electromagnetic interference quite readily
Hence, active filters are highly popular

9. Sallen-Key LP and HP Filters
Sallen-Key active LP & HP filters are extremely popular
Unity Gain Sallen-Key VCVS
Equal-Component Sallen-Key VCVS
Both types use op amp in noninverting configuration as a VCVS
Unity Gain Sallen-Key VCVS 2nd order LP &HP filters
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

10. Unity Gain Sallen-Key VCVS
Most basic active filter with unity gain, second-order
fc for both HP and LP unity gain VCVS with Butterworth response is given by filter
fc = 1/[2πSqrt(C1C2R1R2)]
If α is other than 1.414, appropriate Kf should be included in fc
LP: H(jf) dB = 20 log [1/{Sqrt(1+(α2-2)(f/fc)2+(f/fc)4}]
HP: H(jf) dB = 20 log [1/{Sqrt(1+(α2-2)(fc/f)2+(fc/f)4}]
Normalization: to set α of an LP unity gain VCVS to a desired value and produce a fc of 1 rad/s, we set R1=R2=1 Ω and C1 = 2/α farads, C2 = α/2 farads
Frequency and impedance scaling are used to produce a useful design

11. Unity Gain Sallen-Key VCVS
Impedance Scaling: to scale impedance while maintaining a constant fc, multiply all resistors by the scale factor and divide all capacitors by same scale factor; impedance scaling factor Kz = Znew/Zold
Frequency Scaling: to scale frequency while maintaining a constant impedance, divide all capacitors by frequency scaling OR by multiplying all resistors by scaling factor, while leaving capacitors at a give value; impedance scaling factor Kf = fnew/fold
In order to obtain a useful form of HP unity gain VCVS, set fc to 1 rad/s and capacitors are made equal at 1 farad, while R1 = α/2 and R2 = 2/α

12. Equal-Component Sallen-Key VCVS
Although unity gain VCVS filters get maximum bandwidth form op amp, they are little difficult to design and analyze
Also strict component ratios must be maintained; rather difficult to vary parameters of filter independently
Equal-component Sallen-Key VCVS filters provide quite effective solutions
Designed using equal-values frequency-determining components (R1 = R2 and C1 = C2)

13. LP Equal-Component VCVS+ C1
Vin RB Vo R1 RA R2 C2

14. LP Equal-Component VCVS
Gain of circuit is determined by RA and RB, that are generally not equal
Design of equal-component VCVS requires the gain of op amp to be set at some value that produces desired α
Assuming Butterworth response
fc = 1/[2πSqrt(C1C2R1R2)]
fc = 1/(2πRC) (since R= R1 = R2 , C= C1 = C2)
LP filter may be converted to HP filter with same fc by swapping positions of resistors R1 and R2 with capacitors C1 and C2
Av = 3 – α
RB = RA (2 – α)

15. Second-Order Equal-Component VCVS
Analysis of second-order equal-component VCVS requires a reverse application of design procedure
Determine the passband gain of filter and calculate α; response type is determined by comparing the calculated α with those listed for common filter responses
Apply appropriate frequency scaling factor to fc = 1/(2πRC), and calculate corner frequency of filter

16. Higher-Order LP and HP Filters
Active filters with orders of greater than two are obtained by cascading first- and second-order sections as required
Overall order of a filter that is designed in this manner is equal to the sum of the orders of individual sections used
Obtaining a particular response shape is not quite simple
In order to produce a given response, various sections used to produce the filter must be designed with specific α and fc scaling factors taken into account
When dealing with higher-order filters, all second-order sections used will be of the equal-component VCVS types as
They are easier to analyze and design
LP-HP conversions are performed simply by swapping frequency-determining resistors and capacitors
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

17. Third-Order LP and HP Filters
Designed by connecting first-order RC section to second-order section
Second section will tend to load down first section, producing an overall response that has slightly greater damping than desired
Isolating first section eliminates loading effects of second section
Impedance level of first section should be much lower than that of second section (scaling impedance of first-order section should be 1/10 of impedance level of second section)
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

18. Third-Order Active LP Filters+
Vin RB R1 RA R2 C3 C2 R3 C1
Minimum op amp implementation Vo Vo - +
Vin RB R1 RA R2 C3 C2 R3 C1
Op amp isolation of first section

19. Fourth-Order LP and HP Filters
Designed by cascading two second-order filter sections
Chebyshev filters of order greater than two will exhibit multiple peaks, or ripples in passband; higher the order of filter, more ripples occur
fc of Chebyshev filter is defined as frequency at which ripple channel ends
Passband response for 3-dB LP Chebyshev filters
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

20. Fourth-Order Active LP Filter+
Vin RB RA R1 C2 C1 R2 Vo R3 C4 C3 R4

21. Bandpass (BP) Filters
Active BP and bandstop filters are easily designed
Advantages over passive BP and bandstop filters:
Inductorless design
Ease of tuning and independent parameter adjustment (Q, f0, and BW), electronic control of parameters, and option of adjustable passband gain
Major performance parameter associated with BP and bandstop filters is Q (~ 1-20)
Q is reciprocal of filter α
Since α of BP and bandstop filters is very small, Q is used instead
Q is measure of sharpness of response around filter center frequency f0
Minimum BP filter order is 2
Regardless of Q, slope of curve ultimately approaches a constant value
Second-order one-pole BP normalized response for various Q’s
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

22. Bandpass Filters
BP filters are of even order, with equal ultimate rolloff rates on either side of f0
On semilog graph, response curve will be symmetrical about f0
Continent way of visualizing relationship between order of BP filter and its amplitude response curve is to assume that on each side of f0, ultimate rolloff rate will be that of a HP or LP filter of one-half the order of BP filter
Second-order BP has one pole, fourth-order BP has two poles, and so on
fc (LP section) > fc (HP section)
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

23. Multiple-Feedback BP (MFBP) Filters
Most applications using BP filters requires Q to be higher than unity
MFBP is one-op amp circuit with a second-order, single-pole amplitude response characteristic
Center frequency
f0 = 1/[2πSqrt(C1C2R1R2)]
To ease component selection and to reduce number of variables
C = C1 = C2
For design purposes, values of resistors, based on desired filter characteristics, are determined
R1 = Q/(2πf0AvC)
R2 = Av/(2πf0QC)
Passband Av = -Q Sqrt(R2/R1) - + C1
Vin R2 Vo R1 C2

24. Multiple-Feedback BP (MFBP) Filters
To continuously vary f0 without changing gain or Q, R1 and R2 should be changed at same time keeping the ratio R2/R1 constant; but not practical
Modified MFBP allows this
f0=[Sqrt{(1/R2C2)(1/R1+1/R3)}]/(2π)
R1 = Q/(2πf0AvC)
R2 = Q/(πf0C)
R3 = Q/[2πf0C(2Q2-Av)]
Av = -R2/(2R1)
Av < 2Q2 to obtain finite positive value for R3
f0 of modified MFBP is changed by selecting a new value for R3
R’3 = R3 (fold/fnew)2 - + C1
Vin R2 Vo R1 C2 R3
C = C1 = C2

25. Multiple-Feedback BP (MFBP) Filters+ C1
Vin R2 Vo R1 C2 R3
R3 is replaced with voltage- or current-variable resistor (photocoupler)
Photocoupler is a light-dependent resistor (LDR) encapsulated with a light source
Resistance of LDR decreases as lamp current (an intensity) increases
Varying lamp current varies f0
Electrically adjustable f0 using photocoupler

26. Multiple-Feedback BP (MFBP) Filters
JFET can also be used as voltage-controlled resistor
Negative control voltage applied to gate drives JFET toward pinchoff, increasing drain-to-source resistance
To use JFET effectively, VDS (and input voltage) should be held to a maximum of about 500 mVP-P
For voltages within these limits, JFET will act essentially like a linear resistance whose value is dependent on VGS
Electrically adjustable f0 using a JFET - + C1
Vin R2 Vo R1 C2 R3 G D S Vc
Vc is negative with respect to ground

27. BP Filter Applications
Displays amplitudes of different frequency components that comprise a signal
A sweep oscillator varies f0 of BP and also drives horizontal input of an oscilloscope
Output of BP filter is amplified and rectified, and applied to vertical input of scope
Frequency components that exist in input signal are filtered out at different times during a sweep causing peaks to appear on scope
Horizontal scale of scope represents frequency, while vertical scale represents voltage
Changes in frequency content of input signal are not shown at all points in time
Spectrum Analyzer
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

28. BP Filter Applications
Six-band Graphic Equalizer
Consist of a bank of variable-gain BP filters that are used to boost or attenuate signal components at several fixed frequencies
BP filters are set to various f0 within audio-frequency range
Potentiometers on outputs of filters allow each frequency band to be attenuated or amplified
BP outputs are summed, producing a signal that is tailored to suit operator’s choice
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

29. BP Filter Applications
Filters have Q’s ~ 2 producing identically shaped response curves on semilog graph
Relative low Q is desirable, so that there are no large “holes” or gaps in audio spectrum (20 Hz -20 KHz)
If more filters were used, higher-Q filters could be used
Spectrum is not divided linearly, but rather in a logarithmic manner
Response curves for graphic equalizer
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

30. Bandstop (notch or band-reject) Filters
Used to reject or attenuate undesired frequency components
Bandstop response can be produced by summing outputs of HP and LP filters with overlapping amplitude response curves
Main idea is to set fc for LP filter at a higher frequency than for HP filter
Bandstop response of this circuit makes sense only when phase response curves of HP and LP filters are considered as well as their amplitude response curves
Outputs of HP and LP filters are always out of phase by 180°
Critical point occurs when f = f0 where outputs of filters are equal in amplitude and 180° out of phase resulting in cancellation of signals at output of summer
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

31. Bandstop (notch or band-reject) Filters
Since outputs are being summed and since one filter will be producing output voltage of much grater amplitude for frequencies on either side of f0, two passbands are produced
To produce predictable response, both filters should be of same order with same response shape (usually Butterworth)
Q is determined in same manner as for BP filter
For highest Q, the HP and LP filters should have identical fc
Null frequency is determined by Eq. 6.7
Maximum rejection is ~ 50 dB below passband gain
Overall gain (in dB) at f0 is called null depth; greater the null depth, more effective the filter
Bandstop implemented using second-order equal-component VCVS filters
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

32. Bandstop (notch or band-reject) Filters
Second bandstop filter that relies on cancellation of phase-shifted signals for its response characteristics
Due to inverting gain of MFBP filter, output signal is 180° out of phase with input at f0
When output of MFBP is summed with input signal, it is possible to obtain a bandstop response due to the relative phase inversion of two signals
Null frequency of bandstop is same as f0 for MFBP
To realize maximum null depth, summing amplifier must be designed to compensate for differences between its tow input signals
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

33. State-Variable Filters
Vin R
(LP) Vo - + C
(BP) Vo
(HP) Vo
Rα = R [(3- α)/ α]
Av(BP) = Q
= 1/α

34. State-Variable Filters
State-variable filter is an analog computer that continuously solves a second-order differential equation
State-variable filter produces simultaneous HP, LP, and BP responses
For HP and LP outputs, any practical second-order response shape can be achieved, while for BP output, Q’s of greater than 100 are easily obtained
Can be constructed using three or more op amps
Both integrators use equal-value components, and for convenience, remaining resistors are set equal to integrator resistors or scaled as necessary
fc of HP and LP outputs and f0 of BP output: f0 = 1/(2πRC)
f0 can be varied continuously, without affecting α or Q, by simultaneously varying integrator input resistors while keeping then equal to each other
Passband gain of HP and LP outputs is unity
For BP output, gain at f0 is Av(BP) = Q = 1/α
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

35. State-Variable Filters
Vin R
R/Av
(LP) Vo - + C
(BP) Vo
(HP) Vo
Independently adjustable damping and gain R - +
Av(BP) = AvQ

36. All-Pass Filters
Designed to provide constant gain to signals at all frequencies
Ideally, cover entire frequency spectrum
Flat amplitude response characteristic is quite different from those of other filters (LP, HP, BP, notch)
Produce an output that is shifted in phase relative to input signal
Figure: Output signal leads that of input
For frequencies approaching 0, phase lead approaches 180°
As frequency increases, phase lead of output approaches 0
Phase angle: Ø = 2 tan-1 [1/(2πfR1C1)]
Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

37. All-Pass Filters
Feedback and inverting input resistors must be equal to each other
Absolute values of resistors is not critical, but for minimum offset, parallel equivalent of these two resistors should nearly equal the value of R1
Gain of all-pass filter is unity (a necessary condition for normal operation)
By replacing R1 with a potentiometer (or equivalent voltage-controlled resistance), phase angle of output may be varied continuously
Cascading similar all-pass sections produces and additive phase shift
Two all-pass filters cascaded will approach a maximum phase shift of 360°, three sections will approach a maximum phase shift of 540°,..
Lagging phase angle may be produced by interchanging R1 and C1 with the phase angle: Ø = -2 tan-1 (2πfR1C1)