LOW-PASS FILTERS (3dB frequency, Bode plots) POSITIVE FEEDBACK and OPERATIONAL AMPLIFIERS (Schmitt trigger comparator)

Transcription

1 PH-35 LOW-PASS FILTES (3 frequency, Bode plots POSITIVE FEEDBAK and OPEATIONAL AMPLIFIES (Schmitt trigger comparator I. PUPOSE: This laboratory session pursues two ma objecties. A. La osa First, to build a low-pass filter and measure the put oltage (magnitude and phase as a function of its frequency. A log-log plot (Bode plot of the put signal s frequency will help familiarize with the concepts of 3 breakg pot and decrease of put leels per octae and per decade. Second, familiarize with the functiong of the operational amplifiers, and its application comparator circuits. omparators that uses no feedback and a Schmitt Trigger comparator (that uses positie feedback will be built. II. THEOETIAL ONSIDEATIONS: LOW PASS FILTES II. The scale of decibels omparison of signal amplitudes omparison of Power leels II. Sgle-stage low pass filter Output oltage (magnitude and phase II.3 Bode-plots and the 3 breakpot II.4 Voltage drop at high frequencies ( decibels sgle low-pass filter II.4A oll off per octae II.4B oll off per decade II.5 Dual-stage low pass filter II.. The scale of decibels omparison of signal amplitudes Let s consider two signals of amplitudes A and A respectiely. A The ratio of these two signals is: A Because we often we deal with ratios that change by many order of magnitude durg a testg procedure, the decibel scale is frequently used. By defition, A A log A A ( log, log. 3 log , log log A n n log A, log AB log A log B

2 A Example: A is equialent to +6 A because log 6 A We say, A is +6 relatie to A. Example: A signal A times as large as A is + A signal B one-tenth as large as B is - omparison of Power leels If P and P represent the power of two signal leels, their ratio decibels is defed by, P log P P P ( II. Low pass filter Fig. A low pass filter is obtaed with a resistor and a capacitor connected series while the put is taken across the capacitor. Output oltage (magnitude and phase In Fig., assumg a zero put impedance of the drig source, (which proides, and an fite put impedance of the loadg deice. (to which is connected, we obta, Z Z j j j j e where tan (3 That is, the put oltage lags the put oltage. The ratio of the put and put oltage magnitude is gien by, (4

3 II.3 Bode-plots and the 3 breakpot is the characteristic time response of the low-pass circuit. At = /: The ratio / drops to /. 7 In the decibels scale, this ratio is equal to log.7 = - 3; that is / = -3 The change phase is 45 o. V / V = / [ +( ] / V / V = tan - (- = Log (. Log( Log( Log( / /. Fig. Left: Frequency response of a low pass filter. ight: The right figure displays the same data but a logarithmic scale..7 V / V Log ( V/V Low-pass-Filter_DATA =.5 Bode plot V/V Low-pass-Filter_DATA In general (no just for an filter The frequency at which the put oltage drops by -3 is referred to as the -3 breakpot of a filter (or of any circuit that behaes as a filter.. II.4 Voltage drop at high frequencies ( decibels Let s express the ratio of oltages decibels. From expression (4 we obta, log log log At high frequencies,»/, log large That is, for large log log log (5

4 *Log(V / V *Log(V / V Log (V / V We expect, then, that at large frequencies (i.e. for» /, a plot of / s log should learly decrease with a slope of. This is fact erified the figure below. V / V = / [ +( ] / Low pass fillter = tan - (- F = Log Fig. 3 Lear decrease of the put signal ( decibels at high frequencies. II.4A oll-off per octae We are familiar with the fact that music, one octae the frequency is doubled. (6 Similarly our case of electrical signal changes (as described aboe, we can ask ourseles how much does the oltage ratio expression (5 changes ( decibels per octae? Answer: In one octae (a range where the frequency doubles the horizontal axis Fig. 4 changes by [ log ( - log( ] log.3. Sce the slope is -, a change the horizontal axis by.3 will gie a ertical change equal to - x.3 = -6 Accordgly, we say: In a simple low-pass filter the put oltage drops -6 per octae *Log(V/V Low-pass-Filter_DATA = One octae Log( Log( Fig. 4 oll-off pr octae. Left: Lear drop of the put/put oltage ration ( decibels (7 *Log(V/V Low-pass-Filter_DATA

5 Log (V / V (degrees at high frequencies a logarithmic scale. ight: Same data but a zoomed- scale. Equialently, a simple low-pass filter the put oltage decreases by a factor of per octae. (This follows from the fact that for ( A / A / one has ( A / A log(/ 6. II.4B Decibels per decade In a decade, the frequency changes by a factor of. (8 How much does the oltage ratio expression (5 changes ( decibels per decade? Answer: In one decade (a range where the frequency changes by a factor of the horizontal axis Fig. 5 changes by [ log ( - log( ] log. Sce the slope is -, a change of the horizontal axis will correspond to a ertical axis equal to - x = - Accordgly, we say: In a low-pass filter the put oltage drops - per decade (9 V / V = / [ +( ] / Low pass filter = tan - (- - - E =.5 Octae Decade (degrees -5 Fig. 5 esponse from a sgle-stage low-pass filter 3 6 -

6 Equialently, a simple low-pass filter the put oltage decreases by a factor of per decade. (8 [This follows from the fact that for ( A / A / one has ( A / A log(/.] II.5 Dual-stage low pass filter TASK: Derie an expression for terms of. Fig. 6 Two-stage low-pass filter. III. EXPEIMENTAL ONSIDEATIONS III. FILTE III.A Sgle-stage filter Implement a Low-Pass filter. (You may use =. F and = K; of course, feel free to try other alues. Use a signal generator to proide a susoidal put signal of ~ 4 mv amplitude, and check with the oscilloscope whether the put oltage leads or lags the put oltage. Suggestion: Monitor and the oscilloscope s channels and, respectiely. Make a Bode plot of the magnitude and phase of / as a function of the angular frequency. Make sure to get data oer seeral decades of frequency alues. To measure the phase, use the same technique used Experiment-. Determe experimentally the frequency at which the 3 breakg pot occurs. ompare your experimental results with the expected theoretical alue. Determe experimentally the change of the put signal per octae and per decade. Fig. 7 Low-pass filter.

7 III.B Two-stage low-pass filter. Implement two Low-Pass filters cascade. epeat the same measurement requested part III.A. Fig. 8 Two low-pass filters. Make a Bode plot of the magnitude and phase of / as a function of the angular frequency. Make sure to get data oer seeral decades of frequency alues. To measure the phase, use the same technique used Experiment-. Determe experimentally the frequency at which the 3 breakg pot occurs. ompare your experimental results with the expected theoretical alue. Determe experimentally the change of the put signal per octae and per decade.

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