Basic Filter Theory Review Loading tends to make filters response very droopy, which is quite undesirable To prevent such loading, filter sections may.

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

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

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


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


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

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


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]– For nth order Butterworth response (α = 1.414)

H(jf) dB = 20 log [A/Sqrt(1+(fc/f)2n]

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

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 2nd order LP &HP

filters


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

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

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)

LP Equal-Component VCVS

-

+

C1Vin

RB

VoR1

RA

R2

C2

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 responsefc = 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)

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

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


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)


Third-Order Active LP Filters

Vo

-

+

Vin

RB

R1

RA

R2

C3C2

R3

C1

Vo

-

+

Vin

RB

R1

RA

R2

C3C2

R3

C1

-

+

Minimum op amp implementation

Op amp isolation of first section

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


Fourth-Order Active LP Filter

-

+

Vin

RBRA

R1

C2C1

R2 Vo

-

+

RBRA

R3

C4C3

R4

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

Second-order one-pole BP normalized response for various Q’s

Regardless of Q, slope of curve ultimately approaches a constant value


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)


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 variablesC = 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

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 thisf0=[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 R3R’3 = R3 (fold/fnew)2

-

+

C1

Vin

R2

Vo

R1

C2

R3

C = C1 = C2

Multiple-Feedback BP (MFBP) Filters

-

+

C1

Vin

R2

Vo

R1

C2

R3

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 f0

Multiple-Feedback BP (MFBP) Filters

-

+

C1

Vin

R2

Vo

R1

C2

R3G

D

SVc

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, 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

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

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 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


Six-band Graphic Equalizer

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

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


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

Bandstop (notch or band-reject) Filters

• 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


Second bandstop filter that relies on cancellation of phase-shifted signals for its response characteristics

State-Variable Filters

Vin

RR

(LP) Vo

-

+

R

C

-

+

C

-

+

(BP) Vo

(HP) Vo

RRα = R [(3- α)/ α]

R

R

Av(BP) = Q = 1/α

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


State-Variable Filters

Vin

RR/Av

(LP) Vo

-

+

R

C

-

+

C

-

+

(BP) Vo

(HP) Vo

RR

R

R

-

+

R

Independently adjustable damping and gain

Av(BP) = AvQ

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)]


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)

Basic Filter Theory Review Loading tends to make filters response very droopy, which is quite undesirable To prevent such loading, filter sections may.

Documents

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