University of Pittsburgh

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1 University of Pittsburgh Experiment #1 Lab Report Frequency Response of Operational Amplifiers Submission Date: 05/29/2018 Instructors: Dr. Ahmed Dallal Shangqian Gao Submitted By: Nick Haver & Alex Williams Station #2 ECE 1212: Electronic Circuit Design Laboratory

2 Introduction The purpose of this experiment was to design and evaluate several linear amplifiers using the 741 operational amplifier (op-amp). Special emphasis was placed on each design s frequency response changes in circuit behavior with varying input signal frequency. The 741 op-amp was utilized in two topologies: the inverting amplifier, and the non-inverting amplifier. It was known that both topologies amplify an input signal in a linear fashion, with the inverting amplifier negating, or inverting, the output signal. Amplification of the input signal was measured in terms of gain. Varying gains were achieved by connecting two or more resistors to various points on the op-amp. While this resistor-based design gave plenty of flexibility in terms of gain, it was also known that in any op-amp configuration, the output signal voltage is limited by the +/- DC supply voltage connected to the op-amp. No matter the voltage of the input signal and gain, the output signal will never go beyond the range of the +/- supply voltage. In Experiment #1, the op-amp input signals used were sinusoids of varying frequency. Special attention was given to the op-amp behavior across these varying frequencies. It was known that as frequency is increased beyond a certain threshold, amplifier gain begins to decrease. The frequency at which the gain begins to decrease is referred to as bandwidth. It was known that as gain increases, amplifier bandwidth decreases. This results in lower-gain amplifiers achieving greater bandwidths, while higher-gain amplifiers achieve lesser bandwidths. While op-amp behavior had been studied extensively in both ECE 0031 and ECE 0257, amplifier design and real-world analysis has not. In Experiment #1, we hoped to gain a better understanding of op-amp frequency response, amplifier design based on given criteria, and real-world op-amp analysis. Procedure Part I: The Non-Inverting Amplifier In Part I, two different non-inverting amplifiers were designed, built, and tested one with a voltage gain of A V = 1, and the other with voltage gain A V = 10. In the prelab, schematics were designed for each amplifier. The schematic designed is shown in Fig. 1. Figure 1: Non-Inverting Amplifier Schematic Constructed in Part I During design, the op-amps were assumed to be ideal. The ideal op-amp model says that the op-amp will have infinite input impedance and zero output impedance. Therefore, the currents at both the positive (V + ) and negative (V - ) terminals of the op-amp were assumed to be zero. Also, with zero input current, it can also be assumed that V + is always equal to V -. Using this ideal model, the following input-output voltage relationship can be obtained: G ideal = V o = 1 + R 2 (1) V i R 1 Using this relationship, values of R 1 = R 2 = 0 Ω were selected for the amplifier with A V = 1. This amplifier is commonly referred to as a voltage follower, shown in Fig. 2. The purpose of the voltage follower is to isolate the input voltage source from the output. Output voltage is dictated by the input voltage, while current at the output is only drawn from the supply voltage. For the A V = 10 amplifier, Eq. 1 was utilized to choose values of R 1 = 1 kω and R 2 = 10 kω.

3 Figure 2: Voltage Follower Schematic Constructed in Part I In the prelab, the 3dB bandwidth was predicted for each amplifier. The 3dB bandwidth is defined as the maximum frequency at which the op-amp gain drops by 3dB from the gain at which the frequency is zero. 3dB bandwidth was calculated using the following equation: ω T ω 3dB = 1 + R 2 (2) R 1 In Eq. 2, ω T represents the unity gain frequency, the frequency at which the open-loop gain is equal to one. This frequency is a property of the 741 op-amp and was noted to be approximately 1 MHz according to the 741 datasheet. Using ω T = (2π)(1 MHz) and the resistor values mentioned earlier, ω 3dB was estimated to be 392π krad/s for the A V = 1 amplifier, and 36.8π krad/s for A V = 10. Before constructing the amplifiers, the 741 op-amp was wired using the pinout provided in the datasheet, shown in Fig. 3. The DC power supply was connected to the supply pins for a supply voltage of +/- 15 V. The V + and V - pins were grounded and an output voltage of mv was measured with the multimeter. This voltage is referred to as an offset voltage, resulting from imperfections in IC manufacturing. Under ideal conditions, the output voltage would be zero when the V + and V - terminals are grounded. To counteract this offset, a potentiometer was connected to the offset null pins of the 741. The potentiometer resistance was varied until the output voltage was measured to be zero. Figure 3: 741 Operational Amplifier Pinout Diagram With the op-amp offset voltage corrected, both amplifiers were constructed in accordance with Fig. 1, using the potentiometer to counteract the op-amp offset voltage. The oscilloscope was connected between the 741 output pin and ground, allowing the oscilloscope to act as the load while measuring V o. Input voltage V i was provided by the function generator producing a sinusoid with peak-to-peak voltage V pp = 2.00 V. First, low frequency gain was measured for each amplifier by setting the function generator to a frequency of 1 khz and observing the input and output voltage signals on the oscilloscope, shown in Fig. 4 and Fig. 5. Next, 3dB bandwidth was determined by first calculating the 3dB gain for each amplifier using the following equation: A 3dB = A O (3) 2 Using Eq. 3, the 3dB gain can be calculated given the low-frequency gain A O. Using the measured low-frequency gains, 3dB gains were calculated using Eq. 3, and 3dB bandwidths were determined by varying the input signal frequency with the function generator until the desired gain was measured using the oscilloscope. Low frequency gain, 3dB gain, and 3dB bandwidth are shown in Table 1.

4 Next, the 3dB bandwidths of the amplifiers were used to calculate the op-amp unity-gain bandwidths, f t. The unity-gain bandwidths for each amplifier, as well as the nominal value provided in the 741 datasheet are shown in Table 2. A plot of each amplifier s frequency response was created by varying the input signals using the function generator and recording the amplifier s gains at varying frequencies. Frequency responses for the two amplifiers constructed in Part I are shown in Fig. 6 and Fig. 7. In prelab analysis, op-amp input impedance was assumed to be infinite based on the ideal model. A schematic was designed to measure the impedance of the amplifiers, shown in Fig. 8. Figure 8: Amplifier Input Impedance Measuring Schematic In the schematic in Fig. 8, R 1 was nominally chosen to be 1 kω. The function generator was again set to a sinusoid with V pp = 2.00V and a frequency of 1 khz. To measure amplifier input impedance, V s was first measured using the multimeter. The real value of R 1 was measured with the multimeter. The multimeter was then used to measure V in with the circuit disconnected. This measurement was used to calculate the multimeter input resistance R M. The amplifiers were then connected as shown in Fig. 8 with the multimeter placed in parallel with the amplifier to again measure V in. Measured values for both amplifiers are shown in Table 3 and calculations are shown using Eq. 6. Part II: The Inverting Amplifier In Part II, an inverting amplifier was designed, built, and constructed based on a given set of criteria. The amplifier was to have a 3dB bandwidth up to 20 khz, and a low frequency input impedance of 1 kω. In the prelab, a schematic of the amplifier was designed, shown in Fig. 9. Figure 9: Inverting Amplifier Schematic Constructed in Part II Using the ideal op-amp model, the following input-output voltage relationship can be obtained: G ideal = V o V i = R 2 R 1 (4)

5 To determine the values of R 1 and R 2, unity-gain bandwidth of the amplifier was assumed to be the f t = 128 khz calculated for the voltage follower in Part I. Using Eq. 2, the unity-gain bandwidth, and 3dB bandwidth, R 1 and R 2 were calculated as follows: ω 3dB = ω T 1 + R 2 R 1 (2π)(20000) = (2π)(128000) 1 + R 2 R 1 R 2 R 1 = 5.4 R 2 = 5. 4 kω R 1 = 1 kω Using Eq. 4 and the calculated resistor values, the amplifier gain was calculated to be -5.4 V/V. The amplifier shown in Fig. 9 was then constructed using the resistor values calculated above. As in Part I, the DC power supply provided a supply voltage of +/- 15 V and a potentiometer was used to counteract the offset voltage. Low frequency gain was determined by using the signal generator to supply a sinusoidal input signal with frequency 1 khz with V pp = 2.00 V. Input and output signals were again observed on the oscilloscope, shown in Fig. 10. The 3dB gain was calculated to be V/V using Eq. 3, and the 3dB bandwidth was found to be 35.7 khz using the same procedure as in Part I. This bandwidth was compared to the 20 khz 3dB frequency specified in the amplifier criteria. As in Part I, a plot of the amplifier s frequency response was created by varying the input signals using the function generator and recording the amplifier s gains at varying frequencies, shown in Fig. 11. The inverting amplifier design was then modified to allow the amplifier to compensate for a DC offset in the input signal, while producing an output signal with no DC offset. This design enhancement was accomplished using a potentiometer, shown in the schematic in Fig. 12. Figure 12: Inverting Amplifier with DC Offset Correction with Stabilized Potentiometer The circuit in Fig. 12 negated the DC offset by applying a voltage to the V + pin of the op-amp. This is accomplished using a potentiometer connected to the +/- V CC supply voltage. The resistors shown as R 1 and R 2 in Fig. 12 are used to stabilize the potentiometer selection. These resistor values were chosen to be 200 Ω and 100 Ω, respectively. The circuit was constructed and tested for DC offset levels of 0.5 V and 1.0 V. Input and output signals for these two offset levels were observed on the oscilloscope, shown in Fig. 13 and Fig. 14. Using the schematic shown in Fig. 8, the input impedance of the inverting amplifier was measured. As in Part I, R 1 was nominally chosen to be 1 kω. The function generator was again set to a sinusoid with V pp = 2.00V and a frequency of 1 khz. To measure amplifier input impedance, V s was first measured using the multimeter. The real value of R 1 was measured with the multimeter. The multimeter was then used to measure V in with the circuit disconnected. This measurement was used to calculate the multimeter input resistance R M. The amplifier was then connected as shown in Fig. 8 with the multimeter placed in parallel with the amplifier. Frequencies of V in were varied between 100 Hz and 50 khz. For each input frequency, the corresponding input resistance was calculated, as shown in Fig. 15. Inverting amplifier input impedance can be approximated by the following equation: r i = R 1 + (R i r f ) R 1 + R 2 A d (5) As seen in Fig. 11, amplifier gain decreases with increasing frequency. Therefore, according to Eq. 5, as frequency increases, so does input resistance. This relationship is consistent with the relationship illustrated in Fig. 15.

6 Part III: The Modified Inverting Amplifier In Part III, the amplifier designed in Part II was modified to provide one-half the previous gain while maintaining the same input impedance. In the prelab, the new gain of the op-amp was calculated to be -2.7 V/V. Based on Eq. 2, the 3dB bandwidth was expected to increase with this decreased gain. Based on Eq. 4, new values of R 1 and R 2 were calculated as R 2 = 2.7 kω and R 1 = 1.0 kω. The modified amplifier was constructed, and the function generator was again set to a sinusoid with V pp = 2.00V and a frequency of 1 khz. The input and output signals are shown in Fig. 16. The low frequency gain was measured to be V/V. As in Parts I and II, the 3dB gain was calculated to be V/V using Eq. 3 and the 3dB bandwidth was found using the same procedure as in Part I. As expected, the 3dB bandwidth increased to a measured value of 70.3 khz. As in Part I, a plot of the amplifier s frequency response was created by varying the input signal using the function generator and recording the amplifier s gain at varying frequencies. Frequency response for the modified inverting amplifier are shown in Fig. 17. The 3.3 kω load resistor used thus far was replaced with a decade resistor. For a constant output voltage, decreasing the load resistance increases output current. Due to limited output current, voltage gain can become distorted under conditions in which the required output current is greater than the op-amp s maximum output current. Since the output current of the 741 op-amp is limited to 40 ma, it was predicted in the prelab that voltage gain would become distorted with a load resistance less than 80 Ω, as shown below. Voltage gain was plotted as load resistance was increased, as shown in Fig V = 25 ma R Lmin R Lmin = 80 Ω As can be seen in Fig. 18, the voltage gain of approximately 2.7 V/V remains constant until the load resistance is decreased just below 90 Ω. At this point. Voltage gain decreases linearly as load resistance is decreased as a result of limited output current. Results Part I Fig. 4 and Fig. 5 show the waveform of the noninverting op-amps. Channel 1 (yellow) displays the output of the signal while Channel 2 (blue) displays the input. The first op-amp shows the output is identical to the output. Fig. 5 shows an output increased by a factor of 10, as was designed. The tradeoff of the gain is seen in Figs. 6 and 7, where the operating bandwidth of the op-amp with the 10 V/V gain was one tenth as large. Figure 4: Low-Frequency Input and Output for Non-Inverting Amplifier with Av = 1 V/V

7 Voltage Gain (V/V) Figure 5: Low-Frequency Input and Output for Non-Inverting Amplifier with Av = 10 V/V Non-Inverting Amplifiers Low-Frequency Gain (V/V) dB Gain (V/V) dB Bandwidth (khz) Table 1: Low-Frequency Gain, 3dB Gain, and 3dB Bandwidth for Non-Inverting Amplifiers Non-Inverting Amplifier Unity Gain Bandwidths Av = 1 V/V Amplifier 128 khz Av = 10 V/V Amplifier 9.0 khz 741 Datasheet (Nominal) 1.0 MHz Table 2: Unity Gain Bandwidths of Constructed Amplifiers Compared to 741 Datasheet Nominal Value 1.2 Frequency Response of Non-Inverting Amplifier with Av = 1 V/V Frequency (Hz) Figure 6: Frequency Response of Non-Inverting Amplifier with Av = 1 V/V

8 Voltage Gain (V/V) Frequency Response of Non-Inverting Amplifier with Av = 10 V/V Frequency (Hz) Figure 7: Frequency Response of Non-Inverting Amplifier with Av = 10 V/V Non-Inverting Amplifier Input Impedances Av = 1 V/V Amplifier Av = 10 V/V Amplifier Source Voltage (V S) V V Test Resistance (R 1) MΩ MΩ Meter Voltage (V M) V V Meter Internal Resistance (R M) kω kω Amplifier Input Voltage (V in) V V Amplifier Input Impedance (Rin) MΩ MΩ Table 3: Non-Inverting Amplifier Input Impedance Measurements and Calculations For Av = 1, Rin = For Av = 10, Rin = Rin = Vin R1 Rm (Vs Rm) (Vin (R1 + Rm)) (6) ( ) ( ) ( ( )) = 4.317MΩ ( ) ( ) ( ( )) = 1.402MΩ

9 Voltage Gain (V/V) Part II The inverting op amp can be seen as being phase shifted 90 degrees from the input signal, as shown in Fig. 10. Figs. 13 and 14 show the output of the op-amp after the offset signal was corrected. The wave looks the same as a wave with no DC offset, as was the goal. Fig. 15 shows the increasing input impedance of the circuit as frequency increases. Figure 10: Low-Frequency Input and Output for Inverting Amplifier with Av =-5.4 V/V 6 Frequency Response for Inverting Amplifier with Av = -5.4 V/V Frequency (Hz) Figure 11: Frequency Response of Inverting Amplifier with Av = -5.4 V/V Figure 13: Inverting Amplifier Input and Output with 0.5V DC Offset Correction

10 Rin (Ω) Figure 14: Inverting Amplifier Input and Output with 1.0V DC Offset Correction Input Resistance vs Frequency for Inverting Amplifier Frequency (Hz) Figure 15: Inverting Amplifier Input Resistance vs Frequency Part III Fig. 16 shows the output of the op amp with the reduced gain. It is still phase shifted 90 degrees as it is an inverting op amp. Fig. 17 shows the frequency response of the inverting amp, its operating region being about double that of the op amp from Part II. Fig. 18 shows the effectiveness of the op amp under different loading resistances, it maintains its effectiveness evenly until the load goes below 90 Ω. Figure 16: Low Frequency Input and Output for Modified Inverting Amplifier

11 Voltage Gain (V/V) Voltage Gain (V/V) Frequency Response of Inverting Amplifier with Av = -2.7 V/V Frequency (Hz) Figure 17: Modified Inverting Amplifier Frequency Response for Av = -2.7 V/V Voltage Gain vs Load Resistance for Inverting Amplifier Load Resistance (Ω) Figure 18: Modified Inverting Amplifier Voltage Gain vs Load Resistance Discussion Most of the data collected during Experiment 1 agreed with the pre-lab calculations performed. We found that in most cases, using ideal op-amp analysis produces results that are fairly accurate, and often times the errors from resistor tolerances resulted in greater differences than the errors from using the ideal op-amp model. The first part of the experiment had the greatest difference between calculated and actual results. When initially solving for the 3dB bandwidth, the 741 data chart gave an expected nominal value of 1 MHz, where our experiment for A V = 1 V/V gave a 3dB bandwidth khz. The reason for this discrepancy was that the chart lists the range for a nominal gain, while an op-amp configured as a voltage follower is not increasing voltage, it is still the case that an op-amp configured for a reduced voltage gain. For A V = 0.1 V/V for example, the op-amp would likely reach a 3dB bandwidth closer to the 1MHz bandwidth. It did work out however, that for A V = 10 V/V, the 3dB bandwidth was at 18.4 khz, which was about where expected, as it should be about one tenth the voltage follower s bandwidth. In Part II, we used information from what we learned about the op-amp s physical characteristics to calculate a gain for an amplifier with a 20 khz 3dB bandwidth. Our calculations worked well, as Fig. 11 shows the op-amp functioned neatly up to 20 khz before beginning a harsher drop. The success of this experiment also validated the results of Part I, as the math to calculate the allowed gain

12 drew from there. Next, looking at R in of the inverting amplifier, we saw that the value is at about 1 kω for lower frequencies, which is what was to be expected as R 1 is the same. When frequency was increased, we saw an increase in R in. This was unsurprising, as we know op-amps tend to have lower gains and become less effective at higher frequencies. With Part III, the results matched up very closely to the calculated values. With half the gain as Part II, the 3dB bandwidth was double the bandwidth observed in Part II. The tested minimum load current also matched up with what we expected. Our calculations revealed that 80 Ω should be the minimum load that would still allow maximum gain. Fig. 18 showed that the amplifier had no deviation visible for any of the higher load resistances, and at 90 Ω only the smallest difference occurred, then the amplifier voltage was notably stunted at 80 Ω and below. Overall, the experiment matched most of our initial expectations. Most of the deviation in the data could be explained by resistors variance. A further investigation that would be good follow up would be to look at summing and differential amplifiers to see how well those devices match up to ideal op-amp analysis. Conclusion The first part of the experiment had a large difference between the predicted and experimental 3dB bandwidth. The chart s 1 MHz did not match closely with the measured khz range. However, upon investigation, the chart s value discusses nominal gain which is different from the 1 V/V gain used. Using khz as a reference point for the rest of the experiment in Part I led to all values being in the expected range. The 10 V/V amplifier had a 3dB bandwidth of 18.4 khz, which was in the expected range. The input impedance measurements also differed somewhat from what was expected. The 741 data sheet indicates input impedance of 2 MΩ. The input impedance of an ideal non-inverting amplifier should be infinite, but as this is a physical device, the expectation was to find an input impedance near the device s input impedance. The calculated impedance for the voltage follower setup yielded MΩ, a high value. The calculation to acquire this number used numerous variables, many of which were also calculated from somewhat complicated equations. Small errors in measurement could have grown exponentially as they were referenced many times in order to achieve this calculation. The 10 V/V amplifier did yield MΩ input impedance, but the same calculation was used to yield this number, so it is also in question. Part II of the experimented went as expected. The calculated values all matched with what the test results showed. The potentiometer setup that was used to eliminate the offset voltage worked as expected and was able to fully reduce the offset to 0. Part III also matched closely with the expected values. When the gain was halved, the 3dB range was about doubled. The minimum current that was calculated also matched up with what the experiment showed when using the decade box to modify load resistance. References Dr. Ahmed Dallal s ECE 1212 Lecture Notes Texas Instruments 741 Op-Amp Data Sheet