Analog Circuits and Systems Prof. K Radhakrishna Rao Lecture 21: Filters 1
Review Integrators as building blocks of filters Frequency compensation in negative feedback systems Opamp and LDO frequency compensation Log-Antilog amplifier frequency compensation Analog filters Digital filters 2
What are Filters? Characteristics of an ideal band pass filter Characteristics of an ideal band stop filter 3
What are Filters? Characteristics of an ideal low pass filter Characteristics of an ideal high pass filter 4
What are Filters? Characteristics of an ideal all pass filter Characteristics of a practical filter 5
Where are filters used? FM Receiver Electrocardiograph (ECG) Music systems and hearing aids Graphic equalizer Parametric equalizer 6
FM Receiver Much of the radio transmission is done through frequency modulation (FM) Spectrum allocated for FM transmission is 87.9 MHz to 107.9 MHz Several radio stations operate within this spectrum spaced 200 KHz apart FM 102.9 means the carrier frequency used by this station is 102.9 MHz The radio station has to filter signals outside 102.9 MHz + 75 KHz using a band pass filter before transmitting FM receiver should have a tuned circuit, which is a band pass filter, associated with its antenna to select the station FM 102.9 7
FM Receiver (contd.,) Tuned circuit must have a Q (center frequency/band width) of 686 = 102.9/0.15 Receiver uses an intermediate frequency of 10.7 MHz The local oscillator is adjusted to produce 113.6 MHz (102.9 + 10.7) The mixer produces output in the frequency bands of (10.7+0.075) MHz and (102.9+113.6+0.075) MHz A band pass filter with centre frequency of 10.7 MHz and a band width of 150 KHz is used to select signal 8
Electrocardiograph (ECG) Instrument for recording the electrical activity of the heart ECG electrodes produce signals in the range of 0.05 Hz to 1 khz and 0.1 to 300 mv Interference signals include 50 Hz interference from the power supplies motion artifacts due to patient movement radio frequency interference from electro-surgery equipments defibrillation pulses, pace maker pulses, other monitoring equipment, etc. 9
Electrocardiograph (ECG) (contd.,) Modern ECG will have monitor mode and diagnostic mode. Monitor mode: high pass filter is set at either 0.5 Hz or 1 Hz and the low pass filter is set at 40 Hz. This limits artifacts for routine cardiac rhythm monitoring. High-pass filter helps reduce wandering baseline and the low-pass filter helps reduce 50 Hz power line noise. Diagnostic mode: high-pass filter is set at 0.05 Hz, which allows accurate ST segments to be recorded. The low-pass filter is set to 1000 Hz, in which case a notch filter becomes necessary at 50 Hz. 10
Music systems and hearing aids Necessary to adjust amplification of signals differently in different frequency bands Equalization: Compensates for the acoustical properties of the environment and characteristics of receptor (loud speaker and ear) Equalization may require Low frequency shelf filter (bass level controller): the gain is unity above a certain critical frequency High frequency shelf filter (treble level controller): the gain is unity below a certain critical frequency (shelf frequency) Graphic equalizer 11
Music systems and hearing aids (contd.,) When a low frequency shelf filter is combined with a high frequency shelf filter, it can act as a versatile tone controller. Graphic equalizer Adjusts the relative loudness of audio signals in various frequencies Permits a very detailed control of amplitude vs frequency control Requires several overlapping band pass filters with independent gain controls over these bands Parametric equalizer graphic equalizer which provides independent control over the gain, center frequency, bandwidth, and skirt slopes for each filter. 12
Filters Can be passive electrical networks or active electronic circuits. Historically all filtering functions used to be realized using passive filters using R, L and C With the advent of transistors and integrated circuits there has been requirement for size reduction of filters. This led to the development of active RC filters. LC filters are the most reliable units in the microwave range as the size of the passive components in this frequency range become small. 13
Filter functions Ideal filters (box like behavior) cannot be realized because of requirement of multiple values at the edge of pass band. Electronic circuits can only realize single valued functions. Single valued functions that are approximations to the ideal multi valued filter functions 14
Ideal band pass filtering function δx δx T=1for x0- <x< x 0+ 2 2 δx δx T=0for x<x0- and x>x 0+ 2 2 x = x 0 is the center point of the band and dx is the band width Normalized filter function T=1for-1<X<+1 T=0for X< 1 and X>1 X = x x 0 and (dx/2) = 1. Bandwidth of the normalized function is 2. 15
Ideal normalized function 16
Practical normalized function 17
Physically realizable functions T = N(X) D(X) Order of D(X) must be higher than that of N(X) so as to make the function go to zero as X increases to +. As the function is symmetric around X = 0 it has to be an even function of X. The slope at X = 0 is zero (flatness). A function with these three properties is called a flat function. 18
Physically realizable functions (contd.,) Flat Function T = + 2 + 4 + 2m 0 1 2 m H 1 N X N X.. N X 2 4 2n 1 2 n 1+ K X + K X +.. K X where m < n 19
Maximally flat function A flat function that has all its (n-1) derivatives at X = 0 should be zero This requires N 1 = K 1 ; N 2 = K 2... N m = K m and K m+1 = K m+2... = K n-1 = 0 and K n 0. T will have a value between (1+e 1 ) and (1-e 2 ), where e 1 and e 2 are small (<<1) positive values, within the band defined by -1<X<+1. 20
Maximally Flat Function based on Taylor Series ( ) ( ) 0 ( ) 0 ( ) f x f x f( x) = f( x ) + x x + x x 1! 2! 0 0 0 ( ) ( n ) ( ) 0 ( x x ) 0 ( x x ) f x f x + L 3! n! 3 n 0 0 2 21
Second order maximally flat function e 2 = 0 and e 1 = 0.05, 0.1, 0.2. These Maximally Flat Functions are T = 1 +ε 1 4 1 1+ε X also known as Butterworth functions 22
Butterworth functions First, second and third order for e 1 = 0.1 and e 2 = 0 T T T = = = 1 1 2 1 1+ε X 1 1 4 1 1+ε X 1 +ε +ε +ε 1 6 1 1+ε X 23
Butterworth functions (contd.,) Higher order Butterworth functions have better pass band response and higher rates of attenuation in the stop band. Rate of attenuation close to the edge of the pass band of Butterworth functions may not always be acceptable. 24
Functions to improve response at pass band edge 1 T = 1+ K X + K X 2 4 1 2 + 2 1 2 4 1 2 1 N X T = 1+ K X + K X where K >0 2 where N > K and K > 0 1 1 2 25
Chebyshev Function 1 T = 1+ K X + K X 2 4 1 2 This function can approximate a box like behavior by having K 2 to be positive Choosing K 1 and K 2 to have T =1 at X =1 This requires K 1 = -K 2. T peaks at X 2 = 0.5 and attains a value of (1+K 2 /4) leading to K 2 =4e 1 (1+e 1 ) For a specified variation in the pass band (e 1 ) the parameters of the flat function can be chosen 26
Example e 1 =0.05 and K 2 = 0.19 ; e 1 =0.1 and K 2 =0.0.367 ; e 1 =0.2 and K 2 =0.0.667 Second order Butterworth function with e 1 =0.05 With higher e 1 it is possible to have faster rate of attenuation at the edge of pass band. 27
Chebyshev Function T = ( +ε ) + 2 1 (1 N X ) 1 1 2 4 1 2 1+ K X + K X This function will have value 1+e 1 at X = 0. N 1, K 1 and K 2 are selected to have T=1 at X = 1. K 2 will be positive and N 1 = K 1 for the function to be maximally flat. K 2 = (1+N 1 ) e 1 28
Example Consider positive values of N 1 /K 1 with N 1 = K 1 = 0.5 and 1 e 1 = 0.1 K 2 will then be 0.15 and 0.2. Flat function with numerator polynomial in comparison to second order Butterworth function It is observed that with positive values of N 1 the response of flat function with numerator polynomial is inferior to the second order Butterworth function. 29
Example (contd.,) 30
Inverse Chebyshev Function Negative values of N 1 can make the function go to zero at X>1 if N 1 <1. If e 1 = 0.1 at X =0 and e 2 = 0 at X = 1, then K 2 is positive N 1 = K 1 and K 2 = (N 1 +1) e 1 If the function is to become zero at X = 2, then N 1 = K 1 = -0.25 and K 2 = 0.075. If e 1 = 0.1 at X =0 and e 2 = 0.5 at X = 1, then K 2 is positive, N 1 = K 1 and K 2 = (N+1) e 1. If the function is to become zero at X = 2, then N 1 = K 1 = -0.25 and K 2 = 0.9. 31
Inverse Chebyshev Function (contd.,) The functions with different values of e 2 pass through zero at X =2. The behavior of the function in the stop band beyond X = 2 is better for function with e 2 = 0.5 while the function has much better behavior in the pass band with e 2 = 0. 32
Elliptic Function If K 1 > N 1 the response will slightly peak within the pass band and a better rate of attenuation at the edge of pass band. e 1 = 0.1, N 1 = -0.25 and K 1 = -0.35. K 2 = 0.175 for e 2 = 0 and K 2 =0.675 for e 2 = 0.5. The behavior of the function in the stop band beyond X = 2 is better for function with e 2 = 0.5 while the function has much better behavior in the pass band and at the edge of the pass band with e 2 = 0. 33
Elliptic Function (contd.,) The functions with different values of e 2 pass through zero at X =2. 34
Wideband Using Staggered Narrowbands 35
Conclusion 36