Experiment : Active Filter 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 R network with one capacitor ha the tranfer function () H H() where H i the value of the tranfer function at =, or the dc value, i the natural frequency of the network, and i the complex frequency, + j. For = j have () The ratio of the network tranfer function to it dc value i, therefore, a complex number with a magnitude of (3) And a phae angle of H(j) H j H H(j) ( ) arctan The group delay time, which i commonly referred to a the pule repone time, i d ( ) ( ) d Low Pa Double Pole co ( ) A low pa R network with reitor, two capacitor, and an amplifier ha a tranfer function with a complex conjugate pole pair given by H() H () () (6)
Where / =, the quality factor of the network. For = j the ratio of the tranfer function to it dc value i H(j) H ( ) 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 H() H (8) The band pa function ha the property that H() = H() =. High Pa Double Pole A high pa two pole tranfer function ha the form H() H (9) For the high pa function, H() = while H() = H. Low Pa Double Pole Amplitude Repone Low pa double pole function are characterized by their natural frequency,, and, or a i more common electronic circuit uage, their (=/) or their dampg ratio (=/). Note that critical dampg occur at = (=). ircuit with < (<) are termed underdamped while circuit with > (>) are termed overdamped.
H(j)/H db - =.7 77 = =. =.77 - -3. Figure Amplitude repone of low pa pole filter with a parameter /o 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 H(j H ) 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 below 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
H(j)/H() /(-/ ) / Figure Amplitude repone of low pa pole filter with a parameter.77 Figure Amplitude repone of a low pa two pole filter T Three Specific Filter Type: Butterworth 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 -3dB =. 3 Figure Normalized amplitude repone of pole low pa filter 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. Here, 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 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. 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 ( ) R R RFRF R R R RF R F R F (7) Notice that equation (7) ha the ame form a equation (6). The band pa output at p i given by band ( ) R RRF R R RRF RFR F (8) which ha the ame form a equation (8).
Fally, the high pa output at p i given by high ( ) R R R R R F R R F R F (9) which ha the ame form a equation (9). Filter Deign DO BEFORE OMING TO LAB!!! Before you come to the laboratory you will need to deign a Butterworth, a Beel, and a db hebyhev, each with 3 db corner frequency of khz and D ga of. Ug eqn (7) low() RF RF RRF RF ( ) R () Therefore, the dc ga i imply determed by R. A comparion of equation (7) and (6) how that R FR F So the natural frequency Hz i then given by f R F R F Uually, RF and RF are et equal to one another the determation of the natural frequency. Aga comparg equation (7) and (6) we fd that (3) R (R ) R R RF Sce the dc ga i et by R and the natural frequency i et by RF= RF, the quality factor i et by R. () () 6
Example Determg Natural Frequency, o, for three filter type: 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 = 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. From equation (7), correpondg phae hift at the 3 db frequency of radian/ec are: Butterworth: -9 o, Beel: -7. o, and db hebyhev: -3. o. 7
Lab Experimental Procedure Equipment Lit Active filter circuit prted circuit board Prted circuit board fixture Tektronix MDO 3 Ocillocope Tektronix AFG Arbitrary/Function Generator Prelimary Preparation Before you come to the laboratory you mut complete deign for -pole low-pa Butterworth, Beel, and db hebyhev filter ee p.6 of thi write-up. Deign the filter to have a corner frequency of khz and a D ga of, For thi purpoe you need to calculate value for R, RF = RF, and R for each of the three filter to atify the deign pecification. erify your deign reitance calculation with tructor BEFORE performg procedure tep -3 below. Procedure The prted circuit board with the active filter i hown Figure. All component, with the exception of RF, RF, R, and R 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. R k LF37 RF RF R k k 8 8 - Ground Signal In Band Pa Out + High Pa Out Low Pa Out Figure Active filter circuit prted circuit board howg external connection Do not immediately connect the two dc power upplie to the circuit. Firt, adjut both upplie to + and make appropriate connection to obta +, - and lead to power the circuit chai. Then turn the power upplie off and connect lead to the chai.. Butterworth Filter. Inert the reitance you calculated for R, RF = RF and R to the appropriate ocket on the prted circuit board. To obta accurate reitance value 8
you may need to parallel everal reitor. erify the dc bia connection dicated Figure and then turn the dc upplie on. Make ure you till have proper dc level on jack, 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 Hz to khz 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 Hz quare wave put ufficient to give about peak to peak output, adjut the ocillocope to trigger on the trailg 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) Ue the method illutrated Figure 3 of Lab to obta the frequency repone voltage waveform over the frequency range of Hz to khz. Thi data will erve a plot for the low-pa output at p 8, the band-pa output at p, and the high-pa output at p. Set the function generator amplitude o that the p 8 peak-to-peak voltage i approximately volt maximum. Ue thi amplitude ettg for the p 8, p and p meaurement. Note the maximum and -3dB ga voltage amplitude and the frequencie at which they occur for all cae. Be ure to ave data a format that clearly exhibit ignificant frequencie and amplitude. For tance, appropriately caled creen hot, ug curor, of the LP, BP and HP output or correpondg data dump for each cae might be ued.. Beel Filter. With the dc power upplie off, ert the reitance you calculated for R, RF = RF and R. 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 Hz to khz payg particular attention to the frequency range where the amplitude change i greatet. b) With a 3 Hz 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 R, RF = RF and R. 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 Hz to khz payg particular attention to the frequency range where the output amplitude change rapidly. Here it may be ueful to look at a Bode plot firt. b) With a 3 Hz quare wave put ufficient to give about peak to peak output, record the waveform of the low-pa output at p 8. 9
To Be Included Report Write Up: (a) Tabulate your calculated value of R, RF = RF, and R 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) With both axe appropriately labeled and ug Procedure part c) reult, on eparate graph, preent the voltage v frequency waveform of the low-pa, band-pa, and high-pa output for the Butterworth filter only. ompare the experimental maximum ga for the low-pa, band-pa, and high-pa output to thoe expected theoretically. uantitative, not qualitative, reult are expected here. Give detail of upportg calculation. Reference and Suggeted Readg. 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 Hall, Upper Saddle River, New Jerey, 997) 3. D. E. Johnon, Introduction to Filter Theory, (Prentice-Hall, Englewood liff, 976). J. E. Storer, Paive Network Synthei, (McGraw-Hill, New York, 97).. L. Weberg, Network Analyi and Synthei, (McGraw-Hill, New York, 96). 6. A.B. William, Active Filter Deign, (Artech Houe, Dedham, 97).