Experiment 8: Active Filter October 3, In power circuit filter are implemented with ductor and capacitor to obta the deired filter characteritic. In tegrated electronic circuit, however, it ha not been poible to realize high quality ductor tegrated form, o filter are often implemented with reitor, capacitor, and amplifier. Thee are called active filter. It i alo poible to contruct filter with jut capacitor and amplifier, o-called witched capacitor filter, but we will retrict our experiment to the conventional active filter with reitor, capacitor, and operational amplifier Network Function Low Pa Sgle Pole A low pa network with one capacitor ha the tranfer function where i the value of the tranfer function at, or the dc value, i the natural frequency of the network, and i the complex frequency, σ j. For j have () (j) j The ratio of the network tranfer function to it dc value i, therefore, a complex number with a magnitude of And a phae angle of () (j) φ( ) arctan The group delay time, which i commonly referred to a the pule repone time, i [ φ( ) ] d τ( ) d Low Pa Double Pole co φ( ) A low pa network with reitor, two capacitor, and an amplifier ha a tranfer function with a complex conjugate pole pair given by () α () (3) () () (6)
Where /α, the quality factor of the network. For j the ratio of the tranfer function to it dc value i (j) ( ) j (7) which i a complex number whoe phae angle and pule repone can be determed a equation () and (). Band Pa Double Pole A band pa two pole tranfer function ha the form () The band pa function ha the property that () ( ). α α (8) igh Pa Double Pole A high pa two pole tranfer function ha the form () α For the high pa function, () while ( ). (9) Low Pa Double Pole Amplitude epone Low pa double pole function are characterized by their natural frequency,, and α, or a i more common electronic circuit uage, their (/α) or their dampg ratio ζ (α/). Note that circuit with ζ (α) are ritically Damped. ircuit with ζ< (α<) are termed Underdamped while circuit with ζ> (α>) are termed Overdamped.
(j)/ db - Figure Amplitude repone (ζ.) of a low pa two pole filter. (ζ.8).77 (ζ.77).777 (ζ.877) - -3. / o Figure Amplitude repone of low pa pole filter with a parameter Figure how the magnitude of the ratio of the tranfer function of a low pa two pole filter to it dc value a a function of the ratio of the radian frequency,, to the natural radian frequency,, for everal value of. At the frequency, it i evident from equation (7) that thi ratio i (j ) j / 9 For filter with large value the maximum ga occur at and the maximum ga i. At thi frequency, for example, the amplitude repone of a circuit with would be, decibel, log () db while the amplitude repone of a circuit with.77 would be log (.77) -3 db. More exactly, for /, the maximum ga occur at a frequency le than, namely (-/( )) / and the value of the maximum ga i /(-/( )) /. Figure how a normalized amplitude repone for a two-pole low pa filter. The value of the three frequencie that are of the mot ue two-pole filter deign are o () () 3 () (3) 3
/(-/ ) / (j)/().77 3 Figure Normalized amplitude repone of pole low pa filter Butterworth Filter The implet filter optimization i the Butterworth, or maximally flat amplitude repone filter. Thi filter ha the larget value of that will give an amplitude repone that doe not rie above it dc value. Analyi how that a low pa filter with thi property would have.77. Further, it i traightforward to how that the 3 db bandwidth of the filter i Beel Filter -3dB. () The Beel filter ha the property that it group delay, a defed equation (), i maximally flat, the ame ene a the amplitude repone i maximally flat the Butterworth filter. ere, analyi how that a two pole Beel filter hould have a.77. For the Beel two-pole low pa filter the 3 db frequency i -3dB.78. hebyhev Filter () All two pole low pa filter with >.77 and with no zeroe are properly called hebyhev filter. A hown Figure, they are characterized by a peak the amplitude repone. Thi peak i alo referred to a the pa band ripple and i uually meaured db. For example, a db hebyhev ha a and maximum ripple at very cloe to. The db hebyhev hown Figure ha. and peak at a frequency.777. Workg from equation (7) we can etablih that a db two pole low pa hebyhev filter ha a 3 db frequency uch that -3dB.33. (6)
ircuit Analyi The circuit that we will ue thi experiment i hown Figure 3. Although employg more than one amplifier, it ha low enitivity of the natural frequency and to circuit component value fluctuation and i capable of implementg filter with relatively large. Alo, it provide at eparate output, the low pa, band pa, and high pa tranfer function. k 8 Input k F F 8 k Figure 3 Active filter circuit diagram. P 8 i low pa output, p i band pa output, and p i high pa output. An analyi of the circuit diagram of the active filter Figure 3 how that the ratio of the low pa output voltage at p 8 to the put voltage i given by low ( ) F ( ) F F F F (7) Notice that equation (7) ha the ame form a equation (6). The band pa output at p i given by band ( ) ( ) F F F F (8) which ha the ame form a equation (8).
The high pa output at p i given by high ( ) ( ) F F F (9) which ha the ame form a equation (9). Filter Deign Before you come to the laboratory you will need to deign a Butterworth, a Beel, and a dbhebyhev with a 3 db corner frequency of kz and a D ga of. Ug equation (7) low () () F F F F ( ) Therefore, the dc ga i imply determed by. A comparion of equation (7) and (6) how that F F So the natural frequency z i then given by f π F F Uually, F and F are et equal to one another the determation of the natural frequency. Aga comparg equation (7) and (6) we fd that ( ) (3) F Sce the dc ga i et by and the natural frequency i et by F F, the quality factor i et by. () () Example Butterworth, Beel, and db hebyhev filter are required to have a dc ga of and a 3 db frequency of radian/ec. From equation () we have for the Butterworth cae that 6
radian/ec. From equation () we fd that for the Beel deign,.7 radian/ec while from equation (6) for the db hebyhev we fd that.7 radian/ec. Then from equation (7) correpondg phae hift at the 3 db frequency ( radian/ec) are Butterworth: -9 o, Beel: -7. o, and db hebyhev: -3. o. Experiment Equipment Lit Active filter circuit prted circuit board Prted circuit board fixture P 33A Function Generator P 38A Spectrum Analyzer Prelimary Preparation Before you come to the laboratory you mut complete deign for -pole low-pa Butterworth, Beel, and db hebyhev filter. Deign the filter to have a corner frequency of kz and a D ga of, For thi purpoe you need to calculate value for, F F, and for each of the three filter to atify the deign pecification. Procedure The prted circuit board with the active filter i hown Figure. All component, with the exception of F, F,, and are already on the board. The capacitor denoted a Figure 3 and have been made equal to pf. Input, output, and power upply connection are a hown Figure. There are ocket on the board to allow ertion of the four reitor value. k LF37 F F k k 8 8 - Signal In Ground Band Pa Out igh Pa Out Low Pa Out Figure Active filter circuit prted circuit board howg external connection 7
Do not immediately connect the two dc power upplie to the circuit. Firt, adjut both upplie to, make the appropriate connection to obta and - with the center at ground potential, and turn the power upplie off.. Butterworth Filter. Inert the reitance you calculated for, F F and to the appropriate ocket on the prted circuit board. To obta accurate reitance value you may need to parallel everal reitor. Supply the dc bia connection dicated Figure and then turn the dc upplie on. heck to make ure you till have and -. a) With a uoidal put ufficient to give about peak to peak output, meaure the magnitude and phae of the low-pa voltage tranfer ratio, 8 /, from about z to kz payg particular attention to the frequency range around cut off. Beyond the cut off frequency you may want to creae the amplitude of the put ignal to obta more reliable readg. b) With a 3 z quare wave put ufficient to give about peak to peak output, adjut the ocillocope to trigger on the rig edge of the put pule, and record the waveform of the low-pa output at p 8. Thi ignal i a characterization of the tep repone of the filter. c) onnect the trackg ocillator output of the P 38A pectrum analyzer to the put of the filter. ecord a dimenioned ketche the Bode plot for the low-pa output at p 8, the band-pa output at p, and the high-pa output at p. Be ure to record the maximum ga amplitude for the band-pa output and the high-pa output a compared to the maximum low-pa output. It will be neceary to etablih a reference level on the pectrum analyzer by connectg the trackg ocillator output to the put of the pectrum analyzer.. Beel Filter. With the dc power upplie off, ert the reitance you calculated for, F F and. Now turn the dc upplie on. heck to make ure you till have and -. a) With a uoidal put ufficient to give about peak to peak output, meaure the magnitude and phae of the low-pa voltage tranfer ratio, 8 /, from about z to kz payg particular attention to the frequency range where the amplitude change i greatet. b) With a 3 z quare wave put ufficient to give about peak to peak output, record the waveform of the low-pa output at p 8. 3. hebyhev Filter. With the dc power upplie off, ert the reitance you calculated for, F F and. Now turn the dc upplie on. heck to make ure you till have and -. a) With a uoidal put ufficient to give about peak to peak output, meaure the magnitude and phae of the low-pa voltage tranfer ratio, 8 /, from about z to kz payg particular attention to the frequency range where the output amplitude change rapidly. ere it may be ueful to look at a Bode plot firt. b) With a 3 z quare wave put ufficient to give about peak to peak output, record the waveform of the low-pa output at p 8. 8
eport (a) Tabulate your calculated value of, F F, and for the -pole low-pa Butterworth, Beel, and hebyhev filter. Show your calculation detail. Decribe any difference between thee value and the value actually ued the experiment. (b) On the ame graph plot the low-pa voltage tranfer ratio veru frequency for all three low-pa filter for comparion. Ue a log cale for frequency. On another graph plot the phae hift veru frequency for all three low-pa filter. Dicu the difference thee reult for the three filter. (c) Tabulate the meaured value of 3 db frequency, the phae hift at the 3 db frequency, and the frequency at 9 o phae hift for all three low-pa filter. Dicu how thee value compare with the deign value. (d) Preent and dicu difference the tep repone for the three low-pa filter payg particular attention to any rgg. (e) On the ame graph preent the Bode plot for the low-pa, band-pa, and highpa output for the Butterworth filter. ompare the experimental maximum ga for the lowpa, band-pa, and high-pa output to thoe expected theoretically. uantitative, not qualitative, reult are expected here. Show your calculation detail. eference. Adel S. Sedra and Kenneth Smith, Microelectronic ircuit, th Edition, (Oxford Univerity Pre, New York, New York, ). Alan. Oppenheim and Alan S. Willky, Signal and Sytem, nd Edition, (Prentice all, Upper Saddle iver, New Jerey, 997) 3. D. E. Johnon, Introduction to Filter Theory, (Prentice-all, Englewood liff, 976). J. E. Storer, Paive Network Synthei, (McGraw-ill, New York, 97).. L. Weberg, Network Analyi and Synthei, (McGraw-ill, New York, 96). 6. A.B. William, Active Filter Deign, (Artech oue, Dedham, 97). 9