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Basic Filter Theory Review Loading tends to make filters 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

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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

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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 f c = 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

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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 f c – Critically damped filters: no effect – Underdamped filters: increase in f c – Overdamped filters: decrease in f c Effect of Damping Coefficient on second- order LP LC filter Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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Second-Order LP LC Filters Frequency scaling factors K f indicated relative increase or decrease from f c of an equivalent filter with α = f c of a second-order Butterworth active filter f c = 1/[2πSqrt(C 1 C 2 R 1 R 2 )] If α is changed, new f c would be given by f c = K f /[2πSqrt(C 1 C 2 R 1 R 2 )] 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, K f = Butterworth filter: allowing flattest possible passband; most popular filter – α = 1.414, K f = 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, K f = – 2-dB Chebyshev filters α = 0.895, K f = – 3-dB Chebyshev filters α = 0.767, K f = 1.189

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

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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+(α 2 -2)(f/f c ) 2 +(f/f c ) 4 }] – For n th order Butterworth response (α = 1.414) H(jf) dB = 20 log [A/{Sqrt(1+(f/f c ) 2n }] Second-order HP filters – H(jf) dB = 20 log [A/{Sqrt(1+(α 2 -2)(f c /f) 2 +(f c /f) 4 }] – For n th order Butterworth response (α = 1.414) H(jf) dB = 20 log [A/{Sqrt(1+(f c /f) 2n }]

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Active LP and HP Filters It is not possible to produce passive RC filter with α = 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

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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 Unity Gain Sallen- Key VCVS 2 nd order LP &HP filters Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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

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Unity Gain Sallen-Key VCVS Impedance Scaling: to scale impedance while maintaining a constant f c, multiply all resistors by the scale factor and divide all capacitors by same scale factor; impedance scaling factor K z = Z new /Z old 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 K f = f new /f old In order to obtain a useful form of HP unity gain VCVS, set f c to 1 rad/s and capacitors are made equal at 1 farad, while R 1 = α/2 and R 2 = 2/α

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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 (R 1 = R 2 and C 1 = C 2 )

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LP Equal-Component VCVS -+-+ C1C1 Vin RBRB VoVo R1R1 RARA R2R2 C2C2

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LP Equal-Component VCVS Gain of circuit is determined by R A and R B, 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 f c = 1/[2πSqrt(C 1 C 2 R 1 R 2 )] f c = 1/(2πRC) (since R= R 1 = R 2, C= C 1 = C 2 ) LP filter may be converted to HP filter with same f c by swapping positions of resistors R 1 and R 2 with capacitors C 1 and C 2 A v = 3 – α R B = R A (2 – α)

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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 f c = 1/(2πRC), and calculate corner frequency of filter

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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 f c 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

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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

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Third-Order Active LP Filters VoVo -+-+ Vin RBRB R1R1 RARA R2R2 C3C3C2C2 R3R3 C1C1 VoVo -+-+ RBRB R1R1 RARA R2R2 C3C3C2C2 R3R3 C1C Minimum op amp implementation Op amp isolation of first section

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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 f c 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

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Fourth-Order Active LP Filter -+-+ Vin RBRB RARA R1R1 C2C2C1C1 R2R2VoVo -+-+ RBRBRARA R3R3 C4C4C3C3 R4R4

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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, f 0, 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 f 0 Minimum BP filter order is 2 Second-order one-pole BP normalized response for various Qs Regardless of Q, slope of curve ultimately approaches a constant value Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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Bandpass Filters BP filters are of even order, with equal ultimate rolloff rates on either side of f 0 On semilog graph, response curve will be symmetrical about f 0 Continent way of visualizing relationship between order of BP filter and its amplitude response curve is to assume that on each side of f 0, 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 f c (LP section) > f c (HP section) Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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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 f 0 = 1/[2πSqrt(C 1 C 2 R 1 R 2 )] To ease component selection and to reduce number of variables C = C 1 = C 2 For design purposes, values of resistors, based on desired filter characteristics, are determined R 1 = Q/(2πf 0 A v C) R 2 = A v /(2πf 0 QC) Passband A v = -Q Sqrt(R 2 /R 1 ) -+-+ C1C1 Vin R2R2 VoVo R1R1 C2C2

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Multiple-Feedback BP (MFBP) Filters To continuously vary f 0 without changing gain or Q, R 1 and R 2 should be changed at same time keeping the ratio R 2 /R 1 constant; but not practical Modified MFBP allows this f 0 =[Sqrt{(1/R 2 C 2 )(1/R 1 +1/R 3 )}]/(2π) R 1 = Q/(2πf 0 A v C) R 2 = Q/(πf 0 C) R 3 = Q/[2πf 0 C(2Q 2 -A v )] A v = -R 2 /(2R 1 ) A v < 2Q 2 to obtain finite positive value for R 3 f 0 of modified MFBP is changed by selecting a new value for R3 R 3 = R 3 (f old /f new ) C1C1 Vin R2R2 VoVo R1R1 C2C2 R3R3 C = C1 = C2

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Multiple-Feedback BP (MFBP) Filters -+-+ C1C1 Vin R2R2 VoVo R1R1 C2C2 R3R3 Electrically adjustable f0 using photocoupler 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 f 0

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Multiple-Feedback BP (MFBP) Filters -+-+ C1C1 Vin R2R2 VoVo R1R1 C2C2 R3R3 G D S VcVc Vc is negative with respect to ground Electrically adjustable f0 using a JFET 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, V DS (and input voltage) should be held to a maximum of about 500 mV P-P For voltages within these limits, JFET will act essentially like a linear resistance whose value is dependent on V GS

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BP Filter Applications Displays amplitudes of different frequency components that comprise a signal A sweep oscillator varies f 0 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 Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey Spectrum Analyzer

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BP Filter Applications 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 f 0 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 operators choice Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey Six-band Graphic Equalizer

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BP Filter Applications Filters have Qs ~ 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 Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey Response curves for graphic equalizer

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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 f c 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 = f 0 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

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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 f 0, 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 f c 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 Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey Bandstop implemented using second-order equal- component VCVS filters

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Bandstop (notch or band-reject) Filters Due to inverting gain of MFBP filter, output signal is 180° out of phase with input at f 0 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 f 0 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 Second bandstop filter that relies on cancellation of phase-shifted signals for its response characteristics

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

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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, Qs 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 f c of HP and LP outputs and f 0 of BP output: f 0 = 1/(2πRC) f 0 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 f 0 is A v (BP) = Q = 1/α Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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

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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/(2πfR 1 C 1 )] Operational Amplifiers and Linear Integrated Circuits: Theory and Applications by Denton J. Dailey

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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 R 1 Gain of all-pass filter is unity (a necessary condition for normal operation) By replacing R 1 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 R 1 and C 1 with the phase angle: Ø = -2 tan -1 (2πfR 1 C 1 )

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