Analog Filters D. T A E K T U T U N J I P H I L A D E L P H I A U N I V E S I T Y, J O D A N 2 0 4
Introduction Electrical filters are deigned to eliminate unwanted frequencies Filters can be classified as: Analog or Digital Analog filters can be classified as passive or active and they are implemented using electrical components, such as resistors, capacitors, inductors, and op-amps When op-amps are used, filters become active Digital filters are implemented as programs in computers, microcontrollers, or DSPs
Filter Types: Ideal
ealistic Filters lowpass highpass bandpass bandstop
Cutoff Frequency The passband is the range of frequencies that pass through the filter The stopband is the range of frequencies that do not pass through the filter Cutoff frequency (fc) is the frequency at which the voltage gain of the filter drops to 0.707 (or -3dB) of its maximum value: H( jc ) H 2 max
Lowpass Filter Gain (V/V) Pass band Stop band.0 0.707 Frequency, Hz Cutoff frequency
Capacitors and Inductors Capacitors and Inductors impedances change with the frequency of the signals Capacitors behave as Open Circuit for DC signals Short circuit for high frequencies Inductors behave as Open circuit for high frequencies Short circuit for DC signals Most filters use capacitors (not inductors) because of accuracy, size, and cost
Passive Lowpass Filter V I + _ C + V O _ 0 db -3 db. /C H ( s) Vout Vin / sc / sc Cs H ( ) jc H ( j) 2 ImH ( j ) 2 H ( j) e H ( j) tan Im{ H ( j)} e{ H ( j)} H( j) H ( j) tan ( C) C 2
Passive Lowpass Filter In order to get, f c, we solve the following equation H ( jc ) Hmax 2 2 2 c ( C) 2 c / C fc / 2C 2 ()
Phase (deg) Magnitude (db) MATLAB >> =000; C=.; >> num=; den=[*c ]; >> sys=tf(num,den) Transfer function: --------- 00 s + 0-0 -20-30 Bode Diagram System: sys Frequency (rad/sec): 0.00998 Magnitude (db): -3 >> bode(sys);grid -40 0-45 -90 0-4 0-3 0-2 0-0 0 Frequency (rad/sec)
Amplitude Step esponse Step esponse 0.9 0.8 System: sys ise Time (sec): 220 0.7 0.6 0.5 0.4 0.3 0.2 0. 0 0 00 200 300 400 500 600 Time (sec)
Filter Order Increase Note: By increasing the filter order, the response becomes closer to an ideal filter
Example: LP, HP, BP, BE This circuit demonstrates different filter types By changing the output voltage [ef] Signals and Systems using MATLAB by Luis Chaparro.
Bandpass Filter Example Consider the series LC circuit with output across to design a bandpass filter. Let the poles be at 200 rad/sec and 2000 rad/sec (i.e. cutoff frequencies). The Transfer function becomes H ( s) /( Cs) Ls ( / L) s s^2 ( / L) s /( LC) 2200 s 2200s 2200 s 2200 400000 ( s200)( s2000) s s 200 2000( )( ) 200 2000 2 s s x The last term on the right can be put in Bode form as: 0.0055 jw jw jw ( )( ) 200 2000 eference: University of Tennessee, USA
To: Y() Phase (deg); Magnitude (db) -5 db 0-5 Bode Diagrams From: U() -3 db -0-5 00 50 0-50 -00 0 2 0 3 0 4 Frequency (rad/sec) MATLAB Simulation esult
Active Filters: LP Active filters use op-amps in their design and therefore have gain. An active low pass filter is shown below C Vin - + 2 Vout When input frequency is 0, then the capacitor is open and the circuit becomes a regular inverting op-amp with gain 2/ When input frequency is infinite, the capacitor is short and the Output voltage becomes zero
Active Filters: LP The transfer function, magnitude, and the cutoff frequency of the circuit shown is derived to be C / C ) j H( C ) j / ( Z Z ) j H( c in out 2 2 2 2 2
Active Filters: HP An active high pass filter is shown below C 2 Vin - + Vout The transfer function and the cutoff frequency of the circuit shown is derived to be H ( j) / C c 2 j j / C
Active Filters: Band Pass We can construct an active band pass filter using three cascade stages Active Low Pass Active High Pass Non-inverting Op-amp C L Vin L - + L C H H - + H - + 2 Vout
Active Band Stop Filter C fb V in + _ C 2 2 i + V O _
Active Filter Design Example Problem Statement: A transducer is used as an input to a PC. The measured signal is sinusoidal with high frequency noise added. The signal is shown in figure below. Design an analog signal conditioning circuit to filter out the noise and give a gain of 0 for the sinusoidal signal T signal t shift T noise
Example continued The original signal had a period, T signal, of ms => F signal =,000 Hz The time shift, t shift, is equal to 0.25 ms => phase shift is 0.25ms x360 0 /ms = 45 degrees Then, the signal can be represented as x(t ) 2 ( 000, )t noise 0. 5sin 45 Amplitude frequency Phase shift The noise frequency is calculated to be around 8 KHz And since the signal frequency is KHz => need LP filter with cutoff frequency around 4 KHz
Example continued There are many ways to design this filter. Here is one way: Since the transducer signal is differential => use a difference amplifier at first stage and then active LP filter as second stage V- V+ - + + - - + - + C 2 Vout Stage 0: Buffer Stage : Diff. Amp. Stage 2: LP filter
Example continued Use = KW at the buffers (no gain) The cutoff frequency is c = /C 2 Let C = 0.mF and since c = 2,000 => 2 = 398 W Gain is equal 2 / = 0 => = 39.8 W
Frequency esponse from Poles and Zeros
Consider the filter transfer function: Example
Spectrum Analyzer A Spectrum Analyzer system is used to measure the distribution of power across the frequencies
Conclusion Filters are used to enhance the desired frequencies while eliminating unwanted frequencies. Analog filters are electrical circuits composed mainly of resistors and capacitors (passive) and op-amps (active). Filters have three types: Lowpass, Highpass, Bandpass, and Bandreject