University of Pittsburgh Experiment #6 Lab Report Active Filters and Oscillators Submission Date: 7/9/28 Instructors: Dr. Ahmed Dallal Shangqian Gao Submitted By: Nick Haver & Alex Williams Station #2 ECE 22: Electronic Circuit Design Laboratory
Introduction The purpose of this experiment will be to design, build, and test three active filters: two low pass filters and a band pass filter. The purpose of an active filter is to not only filter out a given range of frequencies, like a passive filter, but also to amplify the signal being filtered. In the case of Experiment #6, amplification was accomplished using the 74 operational amplifier (op-amp). Based on the observations made in Experiment #, known frequency response characteristics of op-amps were considered in the design of the filters in Experiment #6. In addition to the active filters, a Wien-bridge oscillator was also designed, built, and tested. The Wien-bridge oscillator works by utilizing an op-amp (in this case, the 74), and several passive circuit elements to generate a sinusoidal output. It should be noted that, unlike the other circuits designed thus far in the course, the Wien-bridge oscillator requires no input signal to produce the output, relying only on the op-amp supply voltage for power. Procedure Part I: Low-Pass Filters In Part I, the low-pass filter in Fig. was designed given two sets of design constraints. For both cases, resistors R and R 2 were to be equal, and the filter was to be designed to operate at a corner frequency of 38 Hz. Figure : Active Low-Pass Filter Designed in Part I For the first filter, capacitors C and C 2 were required to be equal, and the Q-factor was specified to be.77. Eq. gives an approximation of the Q value to ensure amplifier stability. Using this approximation with the known gain of the non-inverting amplifier, the ratio of R B/R A for Filter was calculated to be.5856. K 3 Q = + R B R A () Next, Eq. 2 was used and time constant RC was calculated given that R = R 2 and C = C 2 for Filter. From this, RC was calculated to be 4.88 x -4. Based on readily available capacitor values, a value of C = C 2 =. µf was chosen, leading to R = R 2 = 48.8 Ω for Filter. f o = (2) 2π R R 2 C C 2 For Filter 2, the amplifier was required to behave as a voltage follower, meaning resistor R B was theoretically zero, while R A was theoretically infinite. Also, the Q factor was specified to be. Given that the only other design criteria for Filter 2 was the corner frequency of 38 Hz, as with Filter, values of R, R 2, C, and C 2 for Filter 2 were chosen to be the same values used in Filter. Design constraints and chosen parameter values for both filters are summarized in Table.
Filter (Calculated) Filter (Actual) Filter 2 (Calculated) Filter 2 (Actual) Corner Frequency (f o) 38 Hz 38 Hz 38 Hz 38 Hz Q-Factor (Q).77.77.. Resistor R A. kω. kω Ω Open Resistor R B 585.6 Ω 59 Ω Ω Short Resistor R 48.8 Ω 47 Ω 48.8 Ω 47 Ω Resistor R 2 48.8 Ω 47 Ω 48.8 Ω 47 Ω Capacitor C. µf. µf. µf. µf Capacitor C 2. µf. µf. µf. µf Table : Calculated and Actual Active Low-Pass Filter Design Parameters for Part I Using the values in Table, the first low-pass filter was constructed and tested. For both Filter and Filter 2, a supply voltage of V CC = ±2 V and a sinusoidal input signal with V pp =. V were used. A frequency response of Filter, shown in Fig. 2, was generated using the function generator, multimeter, and LabView software. Next, the value of resistor R B was altered by ±5% to the values shown in Table 2. R B 59 Ω R B + 5% 69 Ω R B 5% 5 Ω Table 2: Resistor RB with Variation of Approximately ±5% Using the value in Table 2 for R B, two more frequency responses were generated, shown in Fig. 3 and Fig. 4. Next, Filter 2 was constructed with the op-amp functioning as a voltage follower. Again, a supply voltage of V CC = ±2 V and a sinusoidal input signal with V pp =. V were used. The frequency response of Filter 2 is shown in Fig. 5. Part II: Band-Pass Filter In Part II, the band-pass filter in Fig. 6 was designed. The filter was to have a Q-factor of 7, and be designed to operate at a corner frequency of 62 Hz. Using Eq. 3, the ratio of R 2/R was calculated to be 44. Figure 6: Active Band-Pass Filter Designed in Part II Q = 3 + R 2 R (3) Based on readily-available capacitor values, C was decided to be 22 nf. Using this value and the ratio determined in Eq. 3, R was calculated using Eq. 4, allowing R 2 to be calculated based on the ratio established in Eq. 3. f o = 2π C 2 (R R 2 + R 2 ) (4)
A summary of the device parameters chosen for the band pass filter is shown in Table 3. Filter 3 (Calculated) Filter 3 (Actual) Resistor R 555.63 Ω 56 Ω Resistor R 2 244.48 kω 244.7 kω Capacitor C 22 nf 22 nf Table 3: Calculated and Actual Active Band-Pass Filter Design Parameters for Part II Using the parameters in Table 3, the band-pass filter was constructed using a supply voltage of V CC = ±2 V and a sinusoidal input signal of V pp =. V. The input signal voltage was decreased because, unlike the low-pass filter, the active band-pass filter produced a considerably higher gain. If output voltage were to exceed ±2 V, clipping would occur as the op-amp saturated, so the input signal voltage was reduced to prevent this. As in Part I, a frequency response was generated, shown in Fig. 7. As with the first low-pass filter, the value of resistor R 2 was altered by ±% to the values shown in Table 4, and corresponding frequency responses were generated, shown in Fig. 8 and Fig. 9. Part III: Oscillators R 2 244.7 kω R 2 + % 27 kω R 2 % 22 kω Table 4: Resistor R2 with Variation of Approximately ±% In Part III, the Wien-Bridge oscillator in Fig. was designed, constructed, and tested. The purpose of an oscillator is to generate an oscillating (AC) signal from a DC source, in this case, the op-amp supply voltage. Figure : Wien-Bridge Oscillator Designed in Part III The frequency of the Wien-Bridge oscillator is determined by the values of R and C in Fig., while the amplification of the signal is determined by R 2 and R, which function as a non-inverting amplifier. For Experiment 6, values of R = 2 kω and C =. µf were provided. Based on these values, the frequency of oscillation was determined by applying the Barkhausen criterion. Applying the Barkhausen criterion, which requires that the absolute value of L(jω) =, to Eq. 5 shows that the oscillator is unstable under these conditions. V o = A L(jω) (5) Using the provided values of R and C, the frequency of oscillation was determined to be 326.29 Hz using Eq. 6. ω = 2πf = RC (6) Next, the impedance of the parallel (Z p) and series (Z s) combinations of R and C were determined and applied to the Barkhausen criterion considering the frequency determined using Eq. 6. This led to the conclusion that the ratio of R 2/R must be no less than 2 for the Barkhausen criterion to apply for the desired frequency.
Voltage Gain (V/V) Given the required R 2/R ratio, the oscillator was constructed using the provide values of R and C, and a kω resistor, which measured to be 987.2 Ω, for R. A decade potentiometer was connected to function as R 2 and adjusted until oscillations were observed on the output using an oscilloscope. Oscillations were observed, as shown in Fig., at R 2 = 2. kω. Oscillations were observed at a frequency of 35.79 Hz, less than Hz from the predicted value of 326.29 Hz. The value of R 2 was increased until near-square waves were observed, which occurred at R 2 = 2. kω. The near-square waves were observed at a frequency of 64.3 Hz. Next, resistor R was replaced with a lamp, whose resistance varies proportionally with applied voltage. This property can be useful in preventing clipping of the oscillator output. Given a limited op-amp supply voltage, the output waveform can be clipped as its voltage amplitude reaches that of the supply voltage. Considering the ideal gain of the non-inverting amplifier created with R and R 2, increasing R will reduce the voltage gain. This allows the lamp to stabilize the oscillator by varying the resistance of R with output voltage. As output voltage increases, the voltage across the lamp increases, which in turn increases the lamp s resistance. This increase reduces the voltage gain of the amplifier, reducing the output voltage. With the stabilization circuit constructed, decade potentiometer R 2 was again varied until oscillations were observed at the output, which occurred at R 2 = 57 Ω, as shown in Fig. 2. Results Part I Fig. 2, 3, and 4 show the magnitude frequency response of low-pass Filter. Considering the initial value of R B, maximum voltage gain was approximately.6 V/V. Given the values of R A and R B calculated in the prelab, a theoretical voltage gain of.586 V/V was expected, approximately the experimental values observed. A corner frequency of 38 Hz was also predicted, again similar to the experimental corner frequency..8.6.4.2.8.6.4.2 Figure 2: Frequency Response of Filter with RB As can be seen in Fig. 3 and 4, increased voltage gain will result in higher Q-factors. In terms of magnitude frequency response, increased voltage gain will result in steeper frequency response curves and decreased bandwidth.
Voltage Gain (V/V) Voltage Gain (V/V) 2.8.6.4.2.8.6.4.2 Figure 3: Frequency Response of Filter with RB + 5%.6.5.4.3.2. Figure 4: Frequency Response of Filter with RB - 5% Fig. 5 shows the magnitude frequency response of low-pass Filter 2. Maximum voltage gain was approximately V/V, again nearly equal to the experimental voltage gains observed. In the case of Filter 2, the voltage follower configuration limits the voltage gain to approximately V/V. Applying the general voltage gain constraint to ensure filter stability, the equation below can be derived assuming an ideal voltage gain of V/V. ω Q = 4 (7) RC From Eq. 7, to increase Q-factor for a given frequency, the values of R and C would need to be increased due to the inherent fixed voltage gain of the voltage follower.
Voltage Gain (V/V) Voltage Gain (V/V).2.8.6.4.2 Figure 5: Frequency Response of Filter 2 Part II Fig. 7, 8, and 9 show the magnitude frequency response of band-pass Filter 3. In the pre-lab, the maximum gain of the filter was expected to be approximately 5 V/V. The maximum gain of the filter in Fig. 7 was 64.4 V/V. Fig. 7 shows the frequency response for Filter 3 with the initial value of R 2. Fig. 8 shows the response with a smaller resistor, resulting in a lower center frequency as expected. Fig. 9 shows the frequency response using a larger resistance for R 2, and thus shows a higher center frequency. 7 6 5 4 3 2 Figure 7: Frequency Response of Filter 3 with R2
Voltage Gain (V/V) Voltage Gain (V/V) 2 8 6 4 2 Figure 8: Frequency Response of Filter 3 with R2 + % 2 8 6 4 2 Figure 9: Frequency Response of Filter 3 with R2 % Part III Outputs of the Wien-Bridge oscillator constructed in Part III are shown in Fig. and 2. Fig. shows the oscillator output without the stabilizing lamp connected, showing the lowest value of R 2 that generates oscillations. Fig. 2 is the oscillator output with the lamp added for stabilization, and the R 2 value is the highest value before the circuit experiences clipping of its sinusoidal output. Figure : Oscillator Output for Non-Stabilized Circuit with R2 = 2. kω
Figure 2: Oscillator Output for Stabilized Circuit with R2 = 57 Ω Discussion In this experiment, we were able to successfully design, build, and test each part, and get results within the expected range. In Part I, we tested the frequency response of several low pass filters. Here we were met with success as the cutoff frequency was near predicted. The building and testing of the circuit went smoothly, which is mostly what we were expecting. After completing multiple experiments, we have found op-amps to perform near calculated theoretical values, as opposed to MOSFETs and BJTs which usually have more performance issues when implemented. In Part II, we examined band pass filters and implemented several similar circuits. Our circuit performance and results were as expected. We hypothesized that the amplification range would vary from Eq. 4, which did occur. As predicted, a lower R 2 value would both decrease gain and increase corner frequency. In building this circuit we did have some difficulty, with the largest issue being improper capacitor selection due to mislabeling. We identified the issue by manually measuring capacitor values and were able to correct the problem and implement the circuit correctly. Part III proceeded smoothly according to the calculations performed. For the non-stabilized circuit we calculated a resistor ratio of R 2/R = 2 to achieve oscillation, and with a kω resistor and decade potentiometer set to 2.kΩ, oscillations were observed. A possible experiment to follow up this part would be to analyze the performance of the oscillator when loaded and see how well it is able to perform and maintain its oscillation. Conclusion In Part I, the results followed near what was expected, the corner frequency in Fig. 2 was 38 Hz with a maximum gain of.6 V/V, near our expected gain of about.58 V/V. The increase in voltage gain results in an increase in Q-factor, and, as predicted, increased values of R 2 resulted in a steeper drop off in the frequency response curve. The frequency response of our circuit in Part II was very near predicted values, in Fig. 7 the maximum point occurs very near to the predicted 68 Hz. Our maximum gain however, was limited to 64.4 V/V. This might be because of the variance in the capacitor values from the calculations we performed. For our theoretical 22 µf capacitor, 2 µf capacitors, measured to be about 9.5 µf each, were placed in parallel. Part III resulted in oscillation at when R 2 was 2. kω, very near the predicted value of 2 kω. We were surprised at how susceptible to change in resistor values this circuit was, as when the decade box for R 2 was set to 2.9 kω, there was no oscillation or output at all, but a Ω increase resulted in perfect sinusoidal output with a maximum V pp of V. References Dr. Ahmed Dallal s ECE 22 Lecture Notes Texas Instruments ua74 General-Purpose Operational Amplifier Data Sheet