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**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

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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 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

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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 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

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**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

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**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

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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+(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}]

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**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

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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 2nd 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 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

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**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

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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 (R1 = R2 and C1 = C2)

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**LP Equal-Component VCVS**

+ C1 Vin RB Vo R1 RA R2 C2

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**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)

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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 fc = 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 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

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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**

+ 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

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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 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

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**Fourth-Order Active LP Filter**

+ Vin RB RA R1 C2 C1 R2 Vo R3 C4 C3 R4

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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, 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

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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

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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 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

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**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

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**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

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**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

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**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

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**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

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**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

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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 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

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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 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

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**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

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**State-Variable Filters**

Vin R (LP) Vo - + C (BP) Vo (HP) Vo Rα = R [(3- α)/ α] 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, 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

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**State-Variable Filters**

Vin R R/Av (LP) Vo - + C (BP) Vo (HP) Vo Independently adjustable damping and gain R - + 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/(2πfR1C1)] 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 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)

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