Analogue Filter Design

Analogue Filter Design Module: SEA Signals and Telecoms Lecturer: URL: http://www.personal.rdg.ac.uk/~stsgrimb/ email: j.b.grimbleby reading.ac.uk Number of Lectures: 5 Reference text: Design with Operational Amplifiers and Analog Integrated Circuits (3rd edition) Sergio Franco McGraw-Hill 003 ISBN 0070703 School of Systems Engineering - Electronic Engineering Slide

Analogue Filter Design Reference text: Design with Operational Amplifiers and Analog Integrated Circuits (3rd ed) Sergio Franco McGraw-Hill 003 ISBN 0070703 Approx 43 School of Systems Engineering - Electronic Engineering Slide

Analogue Filter Design Syllabus This course of lectures deals with the design of passive and active analogue filters The topics that will be covered include: Frequency-domain filter approximations Filter transformations Passive equally-terminated ladder filters Passive ladder filters from filter design tables Active filters cascade synthesis Component value sensitivity Copying methods Generalised immittance converter (GIC) Inductor simulation using GICs School of Systems Engineering - Electronic Engineering Slide 3

Analogue Filter Design Prerequities You should be familiar with the following topics: SEEA5: Electronic Circuits Circuit analysis using Kirchhoff s Laws Thévenin and Norton's theorems The Superposition Theorem Semiconductor devices Complex impedances Frequency response function gain and phase The infinite-gain approximation SEEC5: Engineering Mathematics School of Systems Engineering - Electronic Engineering Slide 4

Analogue Filter Design Filters are normally used to modify the frequency spectrum of a signal and are therefore specified in the frequency domain: Gain (db) Max stop- band gain Max pass-band gain Min pass-band gain Max stop-band gain Stop-band edge Pass-band edges Stop-band edge log(freq) School of Systems Engineering - Electronic Engineering Slide 5

Analogue Filter Design The design procedure for analogue filters has two distinct stages In the first stage a frequency response function H(jω) is derived which meets the specification In the second stage an electronic ec c circuit cu is designed ed to generate the required frequency response function Filter circuits can be constructed entirely from passive components or can contain active elements such as operational amplifiers School of Systems Engineering - Electronic Engineering Slide 6

Frequency-Domain Filters The design of frequency-domain filters usually starts by deriving a low-pass prototype filter normalised to a cut-off frequency of rad/s This prototype filter is then transformed to the required type and cut-off frequency: Low-pass High-pass Band-pass Band-stop This procedure results in a frequency response function which meets the specification School of Systems Engineering - Electronic Engineering Slide 7

Frequency-Domain Filters An ideal normalised sharp cut-off low-pass filter has a frequency response: Gain H(jω).0 00 0.0 0.0 ω.0 Angular frequency Unfortunately natel such a filter is not realisable and it is necessary to use approximations to the ideal response School of Systems Engineering - Electronic Engineering Slide 8

Frequency-Domain Filters The frequency response function of a circuit containing no distributed elements is a rational function of jω : H (j ω ) a b 0 + a(j ω) + a(jω) +.. + a n (jω 0 + b(j ω) + b(jω) +.. + b n (jω ) ) n n The first stage in the design is to choose suitable values for the order n and the coefficients a 0..a n and b 0..b n Five different approximations will be considered: Butterworth, Chebychev, Inverse Chebychev, Elliptic and Bessel School of Systems Engineering - Electronic Engineering Slide 9

Butterworth Approximation The Butterworth approximation gain is maximally flat That is to say the gain in the pass-band (below ω) isasflat as possible The gain falls off monotonically in both pass-band (below ω) and stop-band (above ω) The gain of a Butterworth filter of order n is give by: H(jω) + ω n School of Systems Engineering - Electronic Engineering Slide 0

Butterworth Approximation Gain (db) 0-3 g s ω ω ω s log(freq) School of Systems Engineering - Electronic Engineering Slide

Butterworth Approximation The poles p k of a Butterworth frequency response function are given by: p x + jy k k k where: x k (k ) sin π n,, K, n k y k ( k cos ) π n Values of k from to n are substituted into this formula giving the n poles School of Systems Engineering - Electronic Engineering Slide

Butterworth Approximation The poles of the Butterworth response p, p,..., p n are then combined to give the Butterworth frequency response function: H(jω) (jω p )(jω p )..(jω p n ) The Butterworth approximation is an all-pole response That is, the numerator consists simply of a constant and there are no zeros School of Systems Engineering - Electronic Engineering Slide 3

Butterworth Approximation 3rd-order Butterworth approximation: p k (k ) sin π 6 + (k ) jcos π 6 k p k 0.5 +j 0.866 0+j00.0 0.0 3 0.5 j 0.866 School of Systems Engineering - Electronic Engineering Slide 4

Butterworth Approximation Poles of H(jω): ) 0.5 + j 0.866 0.5 j 0.866.0 + j 0.0 Combining the poles: H( jω) (jω ((jω) (jω) + 0.5 j0.866)(j ω + 0.5 + j0.866)(j ω 3 + jω +.0)(jω +.0(jω) +.0) +.0jω +.0 +.0) School of Systems Engineering - Electronic Engineering Slide 5

Butterworth Approximation What order n is required to meet specification? Gain (db) 0-3 g s ω ω ω s log(freq) School of Systems Engineering - Electronic Engineering Slide 6

Butterworth Approximation The gain g of a Butterworth filter, expressed in db, is given by: g 0log H(jω) 0log 0 0 (+ ω n 0log 0 ( + ω 0 ) n -/ The stop-band edge is at ω s, and the stop-band gain is required to be less than g s db: ) g g s s n 0log 0 0nlog 0 g s 0log 0 ω s (+ ω ω s n n n n s ) 0log0 ωs School of Systems Engineering - Electronic Engineering Slide 7

Butterworth Approximation Example: Pass-band gain: g p -3 db Stop-band gain: g s -40 db Pass-band edge: ω p.0 Stop-band edge: ω s.5 Using the formula for the filter order: n g s 40.0 0 log 0 ω s 0log 0.5.36 Thus a Butterworth approximation of order is required to meet the specification School of Systems Engineering - Electronic Engineering Slide 8

Butterworth Approximation Response of th-order Butterworth approximation: Gain (db) g p -3 db g s -40 db ω p.0 ω s.5 00 0.0 Frequency (Hz).0 School of Systems Engineering - Electronic Engineering Slide 9

Chebychev Approximation The Chebychev approximation gain oscillates between een 0 db and g p db in the pass-band (below ω) In the stop-band (above ω) the gain falls off monotonically The Chebychev approximation is not a single approximation for each n, but a group of approximations with different values of the pass-band ripple g p Like the Butterworth approximation, the Chebychev approximation is an all-pole response School of Systems Engineering - Electronic Engineering Slide 0

Chebychev Approximation Gain (db) 0 g p g s ω ω ω s log(freq) School of Systems Engineering - Electronic Engineering Slide 5

Chebychev Approximation Response of 6th-order Chebychev approximation: Gain (db) g p -3 db g s -40 db ω p.0 ω s.5 0.0 Frequency (Hz).0 School of Systems Engineering - Electronic Engineering Slide 6

Inverse Chebychev Approx The Inverse Chebychev approximation gain falls off monotonically in the pass-band (below ω) In the stop-band (above ω) the gain at first falls to - db, and then oscillates between - db and g s db The Inverse Chebychev approximation is not a single approximation for each n, but a group of approximations with different values of the stop-band ripple g s The Inverse Chebychev approximation has imaginary zeros School of Systems Engineering - Electronic Engineering Slide 7

Inverse Chebychev Approx Gain (db) 0 g p g s ω ω ω s log(freq) School of Systems Engineering - Electronic Engineering Slide 8

Inverse Chebychev Approx Response of 6th-order Inverse Chebychev approximation: (db) Gain g p -3 db g s -40 db ω p.0 ω s.5 00 0.0 Frequency (Hz).0 School of Systems Engineering - Electronic Engineering Slide 9

Elliptic Approximation The elliptic response gain oscillates between 0 and g p db in the pass-band (below ω) The gain in the stop-band (above ω) oscillates between g s db and - db The elliptic approximation has imaginary zeros For a given filter specification the elliptic approximation gives the lowest order frequency response function School of Systems Engineering - Electronic Engineering Slide 30

Elliptic Approximation Gain (db) 0 g p g s ω ω ω s log(freq) School of Systems Engineering - Electronic Engineering Slide 3

Elliptic Approximation Response of 4th-order elliptic approximation: (db) Gain g p -3 db g s -40 db ω p.0 ω s.5 00 0.0 Frequency (Hz).0 School of Systems Engineering - Electronic Engineering Slide 3

Time-Domain Response Unit-step response of 6th-order Chebychev:.0 ) g(t) 0.0 0s.0s Time 30s 3.0s School of Systems Engineering - Electronic Engineering Slide 33

Bessel Approximation The Bessel approximation is used where a frequency-domain enc filter is required which also has a good time-domain behaviour All filters generate a frequency-dependent phase shift In a Bessel filter the phase varies approximately linearly with frequency and the different frequency components are delayed by the same amount A time-domain waveform is therefore delayed, but is not seriously distorted School of Systems Engineering - Electronic Engineering Slide 34

Bessel Approximation The Bessel approximation has an all-pole frequency response function: a H (j ω ) 0 n b + b (jω) + b (jω) +.. + (j ) In the high-frequency h limit it ω : 0 b n ω a H(j ω ) 0 ω The coefficients are related to Bessel functions: a 0 b 0 b i (n n i b n n i)! i!( n i )! School of Systems Engineering - Electronic Engineering Slide 35

Bessel Approximation Response of 8th-order Bessel approximation: Gain (db) 00 0.0 Frequency (Hz).0 School of Systems Engineering - Electronic Engineering Slide 36

Bessel Approximation Unit-step response of 8th-order Bessel: 0.0 g(t) 0.0 0s.0s Time 30s 3.0s School of Systems Engineering - Electronic Engineering Slide 37

Low-Pass to Low-Pass Transformation This transformation shifts the cut-off frequency to ω 0 : jω jω ω 0 ω.0 ω ω 0 School of Systems Engineering - Electronic Engineering Slide 38

Low-Pass to Low-Pass Transformation Design example: A Chebychev low-pass filter is required with the following specification: Pass-band gain: g p -3dB Stop-band gain: g s -50 db Pass-band edge: f p 000 Hz Stop-band edge: f s 000 Hz This corresponds to a normalised low-pass filter with ω p.0, ω s.0 School of Systems Engineering - Electronic Engineering Slide 39

Low-Pass to Low-Pass Transformation Normalised low-pass filter: Pass-band gain: g p -3dB Stop-band gain: g s -50 db Pass-band edge: ω p.0 0 Stop-band edge: ω s.0 This specification can be met by a 5th-order Chebychev approximation with 3 db pass-band ripple: H (j ω ) (jω) 5 0.0665 + 0.5745( jω) +.45( jω) + 0.5489( j ω ) + 0.4080( j ω ) + 4 3 0.06650665 School of Systems Engineering - Electronic Engineering Slide 40

Low-Pass to Low-Pass Transformation Applying the low-pass to low-pass transformation with ω 0 π 000683: 0.06650665 H(jω) 5 4 3 jω jω jω + 0.5745 +.45 683 683 683.0 0 jω + 0.5489 683 9 +.390 0 (jω ) 8 5 (jω) jω + 0.4080 + 683 0.0665 + 3.686 0 + 6.493 0 6 5 (jω ) 4 (jω) + 0.0665 + 5.705 0 0.0665 (jω ) 3 School of Systems Engineering - Electronic Engineering Slide 4

Low-Pass to Low-Pass Transformation 5th-order Chebychev low-pass with cut-off frequency 000 Hz : Gain (db) Frequency (Hz) 000 School of Systems Engineering - Electronic Engineering Slide 4

Low-Pass to High-Pass Transformation This transformation converts to a high-pass response and shifts the cut-off frequency to ω 0 : j ω ω 0 j ω ω.0 ω ω 0 School of Systems Engineering - Electronic Engineering Slide 43

Low-Pass to High-Pass Transformation Design example: A Chebychev high-pass filter is required with the following specification: Pass-band gain: Stop-band gain: Pass-band edge: Stop-band edge: g p -3 db g s -5 db f p 000 Hz f s 000 Hz This corresponds to a normalised low-pass filter with ω p.0, ω s.0 School of Systems Engineering - Electronic Engineering Slide 44

Low-Pass to High-Pass Transformation Normalised low-pass filter: Pass-band gain: g p -3 db Stop-band gain: g s -5 db Pass-band edge: ω p.0 Stop-band edge: ω s.0 This specification can be met by a 3rd-order Chebychev approximation with 3 db pass-band ripple: H(jω) (jω) 3 + 0.597( jω) 0.506 + 0.984( jω) + 0.506 School of Systems Engineering - Electronic Engineering Slide 45

Low-Pass to High-Pass Transformation H(jω) (jω ) 3 + 0.597( (j ω ) 0.506 + 0.984( (j ω ) + 0.506 Applying the low-pass to high-pass transformation with ω 0 π 000: 0.506 H( (j ω ) 3 566 566 566 + 0.597 + 0.984 + j ω j ω j ω 0.506.984 0 + 9.43 0 7 0.506( jω) 3 (jω ) +.67 0 4 (jω ) + 0.506( (j ω ) 3 School of Systems Engineering - Electronic Engineering Slide 46

Low-Pass to High-Pass Transformation 3rd-order Chebychev high-pass with cut-off frequency 000 Hz : ) n (db) Gain 000 Frequency (Hz) School of Systems Engineering - Electronic Engineering Slide 47

Low-Pass to Band-Pass Transformation This transformation converts to a band-pass response centred on ω 0 with relative bandwidth k : jω ω j ω ω + ω0 k 0 j ω.00 ω ω 0 School of Systems Engineering - Electronic Engineering Slide 48

Low-Pass to Band-Pass Transformation 0th-order Chebychev band-pass centred on f000 Hz with relative bandwidth k 4: Gain (db) 000 Frequency (Hz) School of Systems Engineering - Electronic Engineering Slide 49

Low-Pass to Band-Stop Transformation This transformation converts to a band-stop response centred on ω 0 with relative bandwidth k : jω ω jω k ω + ω 0 j 0 ω.0 ω ω 0 School of Systems Engineering - Electronic Engineering Slide 50

Low-Pass to Band-Stop Transformation 0th-order Chebychev band-stop centred on f000 Hz with relative bandwidth k 4: Gain (db) 000 Frequency (Hz) School of Systems Engineering - Electronic Engineering Slide 5

Passive Filter Realisation Passive filters are usually realised as equally-terminated ladder filters This type of filter has a resistor in series with the input and a resistor of nominally the same value in parallel with the output; all other components are reactive (that is inductors or capacitors) Passive equally-terminated ladder filters are not normally used at frequencies below about 0 khz because they contain inductors School of Systems Engineering - Electronic Engineering Slide 5

Passive Filter Realisation Equally-terminated low-pass all-pole ladder filter: R Input R Output This type of filter is suitable for implementing all-pole designs such as Butterworth, Chebychev and Bessel Order number of capacitors + number of inductors School of Systems Engineering - Electronic Engineering Slide 53

Passive Filter Realisation Alternative equally-terminated low-pass all-pole ladder filter: R Input R Output This type of filter is suitable for implementing all-pole designs such as Butterworth, Chebychev and Bessel Order number of capacitors + number of inductors School of Systems Engineering - Electronic Engineering Slide 54

Passive Filter Realisation Equally-terminated low-pass ladder filter with imaginary zeros in response: R Input R Output This type of filter is suitable for implementing an Inverse Chebychev or Elliptic response School of Systems Engineering - Electronic Engineering Slide 55

Passive Filter Realisation Alternative equally-terminated low-pass ladder filter with imaginary zeros in response: R Input R Output This type of filter is suitable for implementing an Inverse Chebychev or Elliptic response School of Systems Engineering - Electronic Engineering Slide 56

Passive Filter Realisation Low-pass filters can be changed to high-pass filters by replacing the inductors by capacitors, and the capacitors by inductors: R Input R Output This type of filter is suitable for implementing all-pole designs such as Butterworth, Chebychev and Bessel School of Systems Engineering - Electronic Engineering Slide 57

Passive Filter Realisation Low-pass filters can be changed to band-pass filters by replacing the inductors and capacitors by LC combinations C L L C ω 0 C L L C C ω 0 L where ω 0 is the geometric centre frequency of the passband School of Systems Engineering - Electronic Engineering Slide 58

Passive Filter Realisation Low-pass filters can be changed to band-pass filters by replacing the inductors and capacitors by LC combinations R Input R Output This type of filter is suitable for implementing all-pole designs such as Butterworth, Chebychev and Bessel School of Systems Engineering - Electronic Engineering Slide 59

Component Value Determination Suitable component values can be determined by the following procedure:. Select a suitable filter circuit. Obtain its frequency-response function in symbolic form 3. Equate coefficients of the symbolic frequencyresponse function and the required response 4. Solve the simultaneous non-linear equations to obtain the component values School of Systems Engineering - Electronic Engineering Slide 60

Component Value Determination A simpler procedure is to use filter design tables Normalised low-pass Butterworth (R.0 Ω, ω 0.0 rad/s): n C L C3 L4 C5 L6 C7.44.44 3.000.000.000 4 0.765.848.848 0.765 5 0.68.68.000.68 0.68 6 0.58.44.93.93.44 0.58 7 0.445.47.80.000.80.47 0.445 School of Systems Engineering - Electronic Engineering Slide 6

Component Value Determination Design example: a 5th-order low-pass Butterworth filter with cut-off (-3 db) frequency khz Normalised 5th-order Butterworth low-pass filter: Ω.68H.68H Input 0.68F.000F 0.68F Ω Output School of Systems Engineering - Electronic Engineering Slide 6

Component Value Determination Scale impedances: multiply resistor and inductor values by k, divide capacitor values by k Choose k 0000 0kΩ 6.8kH 6.8kH 0kΩ Input 6.8μF 00μF 6.8μF Output School of Systems Engineering - Electronic Engineering Slide 63

Component Value Determination Scale frequency: divide id inductor and capacitor values by k where k π 000.57 0 4 0kΩ.88H.88H 0kΩ Input 4.98nF 5.9nF 4.98nF Output School of Systems Engineering - Electronic Engineering Slide 64

Component Value Determination Response of 5th-order low-pass Butterworth filter: Gain (db) 00 Frequency (Hz) 0000 School of Systems Engineering - Electronic Engineering Slide 65

Component Value Determination Normalised db ripple low-pass Chebychev (R 0Ω.0 Ω, ω 0 0rad/s):.0 n C L C3 L4 C5 L6 C7 0.57 3.3 3.6.088.6 4 0.653 4.4 0.84.535 5.07.8 3.03.8.07 6 0.679 3.873 0.77 4.7 0.969.406 7.04.3 3.47.94 3.47.3.04 School of Systems Engineering - Electronic Engineering Slide 66

Component Value Determination Design example: a 3rd-order high-pass Chebychev filter with db ripple and a cut-off (-3 db) frequency 5 khz Normalised 3rd-order Chebychev high-pass filter: Ω 0.99F.088 Input 0.45H 0.45H Ω Output.6 School of Systems Engineering - Electronic Engineering Slide 67

Component Value Determination Scale impedances: multiply resistor and inductor values by k, divide capacitor values by k Choose k 000 000Ω 99μF 000Ω Input 45H 45H Output School of Systems Engineering - Electronic Engineering Slide 68

Component Value Determination Scale frequency: divide id inductor and capacitor values by k where k π 5000 3.4 0 5 9 5nF 000Ω 9.5nF 000Ω Input 4.36mH 4.36mH Output School of Systems Engineering - Electronic Engineering Slide 69

Component Value Determination Response of 3rd-order high-pass Chebychev filter: Gain (db) k Frequency (Hz) 00k School of Systems Engineering - Electronic Engineering Slide 70

Alternative Configuration Design example: a 3rd-order high-pass Chebychev filter with db ripple and a cut-off (-3 db) frequency 5 khz Normalised 3rd-order Chebychev high-pass filter: 0.45F 0.45F.6 Ω Input 0.99H 09 9 Ω Output.088 School of Systems Engineering - Electronic Engineering Slide 7

Alternative Configuration Scale impedances: multiply resistor and inductor values by k, divide capacitor values by k Choose k 000 000Ω 45μF 45μF 000Ω Input 99H 9 Output School of Systems Engineering - Electronic Engineering Slide 7

Alternative Configuration Scale frequency: divide id inductor and capacitor values by k where k π 5000 3.4 0 5 000Ω 4.36nF 4.36nF 000Ω Input 9.5mH Output School of Systems Engineering - Electronic Engineering Slide 73

Component Value Sensitivity 5th-order 3dB ripple low-pass Chebychev equallyterminated ladder filter with cut-off frequency 000 Hz: ain (db) G ±0% variation of components values Frequency (Hz) 000 School of Systems Engineering - Electronic Engineering Slide 74

Active Filters: Cascade Synthesis The frequency response function is first split into secondorder factors a H(jω) b a b H 0 0 0 + a + b + a + b (jω) + a (j ω ) + b (jω) + a (jω) + b (jω) (j ω ) 0 ( (j ω ) H (j ω )... (jω) (jω) +.. + a +.. + b a b n n 0 0 (jω) (j ω ) + a + b n n (jω) + a (jω) + b (jω) (jω)... Each factor is implemented using a second-order active filter These filters are then connected in cascade School of Systems Engineering - Electronic Engineering Slide 75

Active Filters: Cascade Synthesis If the response to be implemented is derived from an allpole approximation then the second-order factors will be of simple low-pass (H lp ), band-pass (H bp ) or high-pass h (H hp ) form: a H 0 lp (jω) b + b (jω) + b (jω) 0 a(j ω) H bp (jω) b + b (j ω ) + b 0 a(jω) H hp (j ω ) b + b (jω) + b (j ω ) 0 (jω) School of Systems Engineering - Electronic Engineering Slide 76

Sallen-Key Low-Pass Filter C R R Input C Output H lp (jω) p where : + T (jω ) + T T (jω ) R R T R C T R C School of Systems Engineering - Electronic Engineering Slide 77

Resonance Frequency and Q-factor Sallen-Key: Standard response: H lp (jω) + T + H lp (jω) j ω + + ω Q (jω) TT (jω) 0 (j ω ) ω 0 Thus: T T T ω ω Q 0 ω 0 or: ω 0 T Q T T ω T T 0 School of Systems Engineering - Electronic Engineering Slide 78

Sallen-Key Low-Pass Filter Gain(dB) 0 db Q Q 0 0dB Q -0 db - db / octave -40 db 0 00 000 0000 00000 Frequency (rad/s) School of Systems Engineering - Electronic Engineering Slide 79

Sallen-Key High-Pass Filter C C R Input R Output t H hp (jω ) hp where : (jω) T T + T (jω) + T T (jω) C C T R C T R C School of Systems Engineering - Electronic Engineering Slide 80

Sallen-Key High-Pass Filter Gain(dB) 0 db Q Q 0 0dB -0 db Q db / octave -40 db 0 00 000 0000 00000 Frequency (rad/s) School of Systems Engineering - Electronic Engineering Slide 8

Rauch Band-Pass Filter R C C R Input Output T(j ω) H bp (jω) + T (j ω ) + T T (j ω ) where : R R T R C T R C School of Systems Engineering - Electronic Engineering Slide 8

Rauch Band-Pass Filter Gain(dB) 40 db Q 0 0 db Q 5 0 db Q 0 00 000 0000 00000 Frequency (rad/s) School of Systems Engineering - Electronic Engineering Slide 83

All-Pole Cascade Synthesis Design example: 5th-order Chebychev low-pass filter has numerator and denominator coefficients: a 0 0.0665 b 0 0.0665 a 00 0.0 b 6.493 0-5 a 0.0 b.390 0-8 a 3 00 0.0 b 3 5.705 0 - a 4 0.0 b 4 3.686 0-6 a 5 00 0.0 b -9 5.0 0 The frequency response is factored into second order sections by finding the poles (there are no zeros) School of Systems Engineering - Electronic Engineering Slide 84

All-Pole Cascade Synthesis The poles are the roots of the equation obtained by setting the denominator polynomial to zero: -9.0 0 + + j3.75 0 +3-9.0 0 90 0 + - j3.75 0 0 +3-3.463 0 + + j6.070 0 +3-3.463 0 + - j6.070 0 +3 -.5 0 +3 + j0.0 The frequency response is of 5th-order so that the factors are two conjugate pairs of complex roots and a single real root School of Systems Engineering - Electronic Engineering Slide 85

All-Pole Cascade Synthesis Conjugate pole pairs are combined: -9.0 0 + + j3.75 0 +3-9.0 0 90 0 + - j3.75 0 +3 D (j ω ) (j ω + 9.0 0 j3.75 0 ) (jω (jω) + 9.0 0 + (9.0 0 ) + j3.75 0 + (9.0 0 + 9.0 0 )(jω) (jω) +.80 0 + (3.75 0 + 3 3 3 3 ) ) (jω) +.488 0 + 7 School of Systems Engineering - Electronic Engineering Slide 86

All-Pole Cascade Synthesis The denominator of the st-order section: D(jω) (jω p) jω Dividing through by.5 0 +3 : +.5 0 (j ω 4 D ) + 8.966 0 (j ω ) 4 + 3 Frequency response se function of st-order ode filter: H(jω) + RC (j ω ) Thus: RC 8.966 0 4 Let R0 kω; then C89.66 nf. School of Systems Engineering - Electronic Engineering Slide 87

All-Pole Cascade Synthesis Denominator of the st nd-order section: + 3 D(jω) (jω) +.80 0 (jω) +.488 0 Dividing idi through h by.488 0 0 +7 : 4 8 + 7 D ( j ω ) +. 0 (j ω ) + 6.78 0 (j ω ) Frequency response function of nd-order filter: H lp (jω) + T (jω) + T T (jω Thus: T ) 4 5. 0 T 6.054 0 8 3 6.78 0 T.0 TT 0 Let R RR 0 kω; C.0 nf, C 6.054nF School of Systems Engineering - Electronic Engineering Slide 88

All-Pole Cascade Synthesis Denominator of the st nd-order section: + Dividing idi through h by 3.696 0 +7 : D(jω) (jω) + 6.96 0 (jω) + 5 3.696 0 8 D ( j ω ) +.874 0 (j ω ) +.706 0 (j ω ) Frequency response function of nd-order filter: H lp (jω) + T (jω) + T T (jω Thus: T ) + 7 5 6.874 0 T 9.369 0 8 3.706 0 T.888 TT 0 Let R RR 0 kω; C 88.88 nf, C 0.9369nF School of Systems Engineering - Electronic Engineering Slide 89

All-Pole Cascade Synthesis Complete cascade synthesis of Chebychev active filter:.0nf 88.8nF 0kΩ 0kΩ 0kΩ 0kΩ 0kΩ 89.66nF 6.054nF 0.9369nF Input Output School of Systems Engineering - Electronic Engineering Slide 90

General Cascade Synthesis If the response is not derived ed from an all-pole approximation then the second-order factors will be of general form: H (j ω ) a b 0 + a(j ω) + a(jω) 0 + b(j ω) + b(jω) Rauch or Sallen-Key second-order filters are unsuitable and a more complex filter configuration must be used The ring-of-three filter (aka the bi-quad or state-variable filter) can be used School of Systems Engineering - Electronic Engineering Slide 9

Component Value Sensitivity 5th-order Chebychev low-pass implemented as a cascade of active Sallen-Key sections: Gain (db) Frequency (Hz) 000 School of Systems Engineering - Electronic Engineering Slide 9

Copying Methods Copying methods are ways of designing active filters with the same low sensitivity properties of passive equally- terminated ladder filters The starting point for all copying methods is a prototype passive filter This is then copied in some way which preserves the desirable properties p of the passive filters but which eliminates the inductors The copying method that will be described here is based on inductor simulation School of Systems Engineering - Electronic Engineering Slide 93

Positive Immittance Converters V I V I V 3 4 3 V V + I 3 V 3 V3 + V V + 3 + 3 V V 0 V 0 + V 0 4 School of Systems Engineering - Electronic Engineering Slide 94

Positive Immittance Converters 4 3 I V V + Positive Immittance Converters 3 3 3 3 V V V + + + 0 3 3 V V + + + Substitute to remove V and V 3 : 0 + 3 3 3 3 ) ( V V V + + 0 0 3 3 ) ( V V + + 0 3 4 0 ) ( ) ( V I V + + School of Systems Engineering - Electronic Engineering Slide 95 0

Positive Immittance Converters Multiply both sides by : Positive Immittance Converters Multiply both sides by 0 : ) ( 4 0 0 3 0 0 I V V + ) ( ) ( 3 0 3 4 0 0 3 0 0 V I V V + + + Remove cancelling terms: The impedance of the PIC is given by: 4 0 3 I V The impedance i of the PIC is given by: 4 0 V 3 4 0 I V i School of Systems Engineering - Electronic Engineering Slide 96

Inductor Simulation Let 0,,, and 4 be resistors of value R, and 3 be a capacitor of value C: or: i 3 04 R j 3 R / jωc i jωl L where L CR ωcr The PIC therefore simulates a grounded inductor of value CR By correct choice of R and C the PIC can be made to simulate any required inductance. School of Systems Engineering - Electronic Engineering Slide 97

Copying Methods 5th-order Chebychev high-pass passive filter: 0kΩ 9.34nF 7.05nF 9.34nF Input 4.78H 4.78H 0kΩ Output Let R0 kω; then: CR 8 4.78H C 4.78 0 F School of Systems Engineering - Electronic Engineering Slide 98

Copying Methods 5th-order Chebychev high-pass active filter: 0kΩ 9.34nF 7.05nF 9.34nF 0kΩ 4.78nF 0kΩ 4.78nF Inp put 0kΩ 0kΩ 0kΩ 0kΩ 0kΩ 0kΩ 0kΩ Ou utput School of Systems Engineering - Electronic Engineering Slide 99

Copying Methods 3th-order elliptic high-pass passive filter: kω 38nF 395nF Input 85nF.9H kω Output Let R kω; then: CR 6.9H C.9 0 F School of Systems Engineering - Electronic Engineering Slide 00

Copying Methods 3th-order elliptic high-pass active filter: kω 38nF kω.9μf 85nF 395nF Input kω Output kω kω kω School of Systems Engineering - Electronic Engineering Slide 0

Analogue Filter Design J. B. Grimbleby, 9 February 009 School of Systems Engineering - Electronic Engineering Slide 0