EXPERIMENT 6 LOW-PASS FILTERS, 3dB BREAKPOINT and COMPARATORS

Transcription

1 PH-35. La osa EXPEIMENT 6 LOW-PSS FILTES, 3 BEKPOINT and OMPTOS I. PUPOSE: This laboratory session pursues two ma objecties. First, to build a low-pass filter and measure the put oltage (magnitude and phase) as a function of its frequency. 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. THEOETIL ONSIDETIONS 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.4 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 and respectiely. The ratio of these two signals is: 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, log () log, log. 3 log , log log n n log, log B log log B

2 Example: is equialent to +6 because log 6 We say, is +6 relatie to. Example: signal times as large as is + 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. 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. t = /: 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 = / [ +( ) ] / = tan - (- ) Log () Bode plot -4 = = 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. / V 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, / V log log log t high frequencies,»/, log large That is, for large log log log (5)

4 We expect, then, that at large frequencies (i.e. for» /), a plot of log / s should learly decrease with a slope of. This is fact erified the figure below. Log ( / V ) / V = / [ +( ) ] / Low pass fillter Phase =.5 = tan - (- ) Phase Log Fig. 3 Lear decrease of the put signal ( decibels) at high frequencies. II.4 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? nswer: In one octae (a range where the frequency doubles) the horizontal axis Fig. 4 changes by ( - log ( log.3. [ log ] Sce the slope is -, a change the horizontal axis by.3 will gie a ertical change equal to - x.3 = -6 ccordgly, we say: In a simple low-pass filter the put oltage drops -6 per octae *Log(V / V ) = Log() *Log( / V ) One octae.5.5 Log() Fig. 4 oll-off pr octae. Left: Lear drop of the put/put oltage ration ( decibels) at high frequencies a logarithmic scale. ight: Same data but a zoomed- scale..3 (7)

5 Equialently, a simple low-pass filter the put oltage decreases by a factor of per octae. (8) (This follows from the fact that for ( / ) / one has / ) 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? nswer: 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 = - ccordgly, we say: In a low-pass filter the put oltage drops - per decade (9) / V = / [ +( ) ] / Low pass filter = tan - (- ) Log ( / V ) =.5 Phase 6 3 Octae Decade Phase(degrees) -5 - Fig. 5 esponse from a sgle-stage low-pass filter

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 ( / ) / one has / ) log (/).) ( II.5 Dual-stage low pass filter TSK: Derie an expression for terms of. Fig. 6 Two-stage low-pass filter. III. EXPEIMENTL ONSIDETIONS III. FILTE III. 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. III.B Two-stage low-pass filter. Fig. 7 Low-pass filter.

7 Implement two Low-Pass filters cascade. epeat the same measurement requested part III.. 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. III. OPETIONL MPLIFIES and OMPTOS III. omparator (with no feedback) III.B omparator with positie feedback: Schmitt trigger Op amps are widely used to amplify dc or ac signals. In this lab session, we will use LM358P whose termal connections are shown Fig. 9. = ( (+) - (-) ) is referred as the open loop ga of the opamp, which general depends on the frequency, when ac-signals are used at the put. (-) (+) +V Positie supply - + -V Negatie supply +V -V / +V / ( + - ) -V -V V / = V For V = V and = 5 Fig. 9 Left: Op-amp termals. enter: P assignment for the dual op-amp LM358P. Use Vcc=V. ight: Input/put olatge characteristics.

8 (The dependence of with frequency will be addressed Lab #7, the context of op-amps applications with negatie feedback.) See also the companion notes Operational mplifier posted the website of our course, PPLITIONS: OMPTOS and SHMITT TIGGE Many applications require knowg which or two signals is larger, upon which a termal may need to be actiated (or deactiated). In a digital oltmeter, for example, order to conert an analog signal to a digital number, the unknown oltage is applied to one put of a comparator, while a lear ramp is applied to the other put. digital counter counts the cycles of a clock while the ramp is less than the unknown oltage and display the result when both signals are equal. The resultant count is proportional to the unknown oltage. In this session we will build first a simpler comparator and then a more reliable Schmitt Trigger. III. omparator (with no feedback) The circuit shown the figure compares two oltages (V and V + ) and amplifies the difference between them. The resistors and establish the switchg pot at which the put will change state (V to -V, or ice ersa.) Set up such a circuit usg resistors the K to K range, determe the trigger pot oltage, and exame the leels at the put and puts. Sketch a plot of s V. How would you use that circuit to measure an unknown resistor? V V Fig. omparator circuit. +V -V or ground omparator with positie feedback: Schmitt trigger The simple comparator presents disadantage when the put oltage is noisy, causg the put to make seeral transitions as the put passes through the trigger pot. This problem can be soled by usg positie feedback. The effect of and is to make the circuit hae two thresholds, dependg of the put state. Set up such a circuit usg resistors the K to 3 K range, determe the trigger pot oltage, and exame the leels at the put and puts. Sketch a plot of s V. Verify that the put depends both on the put oltage and on its recent history, an effect called hysteresis.

9 +V V - + V -V Fig. Schmitt trigger circuit. Notice a fraction of the put oltage feedbacks to the positie put of the operational amplifier.