Operational Amplifier Joshua Webster Partners: Billy Day & Josh Kendrick PHY 3802L 10/16/2013
Abstract: The purpose of this lab is to provide insight about operational amplifiers and to understand the role they play in circuitry. In this lab, a circuit is constructed with an operational amplifier being used as a summing amplifier. Data is recorded for different individual experiments with the circuit. The first experiment is to find what effect grounding the inputs has on the output voltage and gain. The second experiment shows the effects of solely increasing the input voltage, and allows us to calculate the theoretical (nominal) values for the output voltage. The third experiment is to show that changing the supply voltage affects the saturation value of the output. The fourth experiment shows the effect that the frequency has on gain. The fifth experiment is to show how the slew rate affects the output signal wave form. The sixth and final part of the experiment is to measure the rise time and calculate the slew rate. The slew rate of the operational amplifier was determined to be 0.536 V/μs. 1
Table of Contents Abstract:... 1 Introduction... 3 Background... 4 Experimental Techniques... 7 Diagrams and Images... 7 Data... 11 Part 1:... 11 Parts 2 & 3:... 11 Part 4:... 12 Part 5:... 13 Part 6:... 14 Part 7:... 15 Part 8:... 18 Analysis... 19 Discussion... 22 Conclusion... 23 Appendix... 24 References... 25 2
Introduction An operational amplifier or op amp is a device that is used extensively in analog electric circuitry. Op amps can be utilized for a number of mathematical tasks including addition, subtraction, multiplication, division, differentiation, and integration. It has dual-inputs, a singleoutput, and functions as a linear amplifier with a high open-loop gain, high input resistances, and low output resistance. 1 Since the resistances of the inputs are very high, the current running through an op amp is very small, which allows us to take the input current to be zero. This experiment deals with the normal operation of an op amp, in which we will be supplying symmetric voltages, and using it as a differential amplifier. A differential amplifier amplifies the voltage difference between two input signals. The op amp chip being utilized is a 741A op amp, which comes in an 8-pin dual-inline package (DIP). 2 The following main sections of this report will consist of the background, experimental techniques, data, analysis, discussion, and a conclusion. Also included will be an appendix for any extra information that doesn t necessarily fit into the other sections, and a section for any references made in the text. 1 (Operational Amplifiers (Op Amps), 2001) 2 (PHY 3802L Experiments) 3
Background Operating in normal mode, we will apply symmetric supply voltages across the operational amplifier. The golden op amp rules state that the input and output currents are zero, and the input and output voltage difference is zero. These rules, however, are used as a close approximation to reality and are only entirely true for an ideal op amp. The output voltage can be represented by, ( ) ( ) In the above equation: V out is the output voltage, A 0 is the open-loop gain, and V + and V - are the positive and negative input voltages respectively. Since op amps are used in circuits with negative feedback, the effective input voltage represented as: can be In the above equation, B is the feedback factor, which is determined by the feedback circuit. The amplification with feedback, also known as closed-loop gain is described using the following equation: ( ) ( ) In the above equation: A f is the amplification with feedback or gain, V out is the output voltage, and V in is the input voltage. From the property of the op amp, we find: Arranging terms, ( ) ( ) ( ) Where A 0 is the open loop gain and B is the feedback factor. This expression shows that the closed loop gain is smaller than the open loop gain. If then. This means that the gain of the amplifier is only dependent on the feedback factor B, and not on the open loop gain A 0. The variation of A 0 is insignificant. The uncertainty in the gain can be calculated using the following equation with the average uncertainties in the measurements: 4
( ) Using the fact that the circuit is a scaling summer circuit 3 and Ohm s Law, an equation for V out can be obtained. ( ) V out is the output voltage, V A and V B are the input voltages, and R f, R 1, and R 2 are the resistances of the resistors. To find the uncertainty in V out we must use the error propagation formula found in the Appendix (A.1): ( ) ( ) ( ) ( ) From this, we find: ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) The uncertainties in the resistor values are given by the individual resistors. The slew rate of an op amp can be defined as the slope of the output voltage versus time: An equation can be formulated that relates the slew rate to the frequency: ( ) 3 (Carter & Brown, 2001) 5
( ) ( ) 4 For the equation above: is the output voltage. is the frequency, one cycle is seconds with a period of 2π, and V out Solving equation 10 for the frequency allows us to estimate the frequency at which gain begins to decrease. Keep in mind that this equation is for a slew rate in units of V/s, and that this equation is used as an estimate for the frequency. ( ) 4 (Research Solutions and Resources LLC, 2011) 6
Experimental Techniques Diagrams and Images Diagram 1: This diagram shows the color coding for resistors. It was used to determine which resistors we needed for our circuit. This image is from the lab website. Figures 1 & 2: Shown below is the operational amplifier setup that was used in this lab. Amplifiers are often represented as a triangle in diagrams. Using the op amp as a differential amplifier; pins 2 and 3 are for the input signals, pin 4 is for the negative supply voltage, pin 7 is for the positive supply voltage, and pin 6 is for the output. These images are from the lab manual. 7
Figure 3: Shown below is the circuit constructed for this lab. v B and v A are inputs 1 and 2 respectively, R 1 and R 2 are input resistors, R f is a feedback resistor. This image is from the lab manual. Image 1: This image shows the actual circuit that was constructed for this experiment. R f 741A Op Amp R 2 R 1 8
Image 2: The image below shows the oscilloscope (top, Tektronix TDS2012B), and function generator/power supply (bottom, FG 501A 2MHz Function Generator, DM 502A Autoranging DMM, PS503A Dual Power Supply). The summing amplifier shown in Figure 3 was constructed as shown in Image 1. A single 2 kω resistor was not available, so two 1 kω resistors were used in its place. Part 1 in the Data section shows the values of the individual resistors. A 741A op amp with a dual-inline package was used. The circuitry was connected to the supply voltage and function generator device, and the supply voltage was set to +12 V and -12 V. Using the function generator, a 1 khz sinusoidal signal of about 100 mv was applied to one input with the other grounded. The sign and magnitude of the amplification factor were recorded. This was repeated, with the signal and ground connected to their respective opposite inputs as before. Next, the same signal was connected to both inputs. The output magnitude was measured and compared to predicted values. The amplitude of the input signal was to be varied starting from 100 mv going up to 1.2 V, however, a systematic error due to human error in recording the measurements of the device caused the readings to be off by one-half of their intended value. At each increment, the output voltage amplitude and gain were determined. A plot was then constructed after the data was obtained and the nominal output amplitude was determined using equation 6. Then supply voltage was changed to ±10 V, and the same sets of measurements were recorded at the same input signal amplitudes. 9
The supply voltage was set back to ±12 V. The gain as a function of frequency was measured for an input amplitude of around 500 mv (actually 250 mv, because of a systematic error). The data for 3 frequency values per decade was recorded. For example, the three frequency values for the first decade would be at intervals around 100 Hz, 300 Hz, and 500 Hz. The sinusoidal input signal was then changed to a square input signal. For low, middle, and high frequency values, the output signals were observed and recorded in the form of images. The input signal was then changed to a triangular wave, and the same data was recorded. Finally, the frequency was set to 1 khz with an amplitude of 500 mv (250 mv). The rise time of the square wave signal was determined using the oscilloscope for both the input to the amplifier and the output from the amplifier. The rise time can only accurately be found by zooming in with the oscilloscope until the trough and peak of the input or output takes up the entire screen (graph). This also allows for the rise time to be estimated by eye using the graph itself. 10
Data Part 1: Resistor values are R 1 =R 1a + R 1b = (1052 ± 1 Ω) + (1063 ± 1 Ω) = 2115 ± 2 Ω, R 2 = 1056 ± 1 Ω, and R f = 10120 ± 10 Ω. R 1 is the sum of two resistors. Parts 2 & 3: Table 1: With the circuit constructed, a 1.014 khz sinusoidal signal was applied at 102 mv with input one (V A or R 1 ) grounded. The resultant output, input 1 grounded V out, was a sinusoidal wave at 980 mv. A signal of 1.018 khz at 103 mv with input 2 (V B or R 2 ) grounded produced an output signal, input 2 grounded V out, of 1.016 khz at 488 mv. With the same signal of 1.016 khz at 100 mv applied to both inputs the output was 1.021 khz at 1.420 V. Freq. (Hz) V in V in V out V out Gain (A f ) Input 1 Grounded 1014 102 1 980 10 9.61 Input 2 Grounded 1018 103 1 488 1 4.74 Both Inputs Same Signal 1016 100 1 1420 10 14.20 11
V out Webster Part 4: Table 2: This table shows the values obtained through the measurements taken with the supply voltage set to ±12 V and their uncertainties due to the accuracy of the device. V in and V out are the input and output voltages respectively, V in and V out are the uncertainties in the input and output voltages respectively. The nominal output was calculated using equation 7 and the uncertainty in the nominal output, σ Nominal, is calculated using equation 8. The nominal voltage was taken to be the absolute value, for means of comparison and graphing. V in (V) V in V out V out Freq. (Hz) Gain Gain Nominal Output σ Nominal 51 1 720 10 990.1 14.1 0.20 733 10.74 101 1 1440 10 990.1 14.3 0.10 1451 10.81 148 1 2140 10 990.1 14.5 0.07 2126 10.92 200 1 2820 10 989.1 14.1 0.05 2874 11.08 250 1 3540 10 990.1 14.2 0.04 3592 11.28 302 1 4320 10 990.1 14.3 0.03 4339 11.54 344 1 4920 10 990.1 14.3 0.03 4943 11.77 400 1 5650 100 989.1 14.1 0.25 5747 12.12 452 1 6400 100 990.1 14.2 0.22 6494 12.49 515 10 7350 100 990.1 14.3 0.19 7400 107.36 550 10 7900 100 989.1 14.4 0.18 7903 107.40 600 10 8600 100 988.1 14.3 0.17 8621 107.45 Graph 1: A plot of the output amplitude as a function of the nominal output amplitude. Errors bars were included, but are very small compared to the scale of the graph. The errors are listed in Table 2. The absolute value of the nominal output amplitude is shown in this graph. 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 V out vs V in (Part 4) 0 100 200 300 400 500 600 700 V in Series1 12
Nominal Output Voltage Webster Part 5: Table 3: This table shows the values obtained through the measurements taken when the supply voltage was set to ±10 V. The data headings are the same as in Table 2 and are explained there. V in V in V out V out Freq. (Hz) Gain Nominal Output 51 1 710 10 987.5 13.9 733 103 1 1460 10 990.1 14.2 1480 150 1 2160 10 990.1 14.4 2155 200 1 2860 10 990.1 14.3 2874 250 1 3540 10 988.1 14.2 3592 302 1 4320 10 988.1 14.3 4339 352 1 5100 100 989.1 14.5 5058 400 1 5650 100 989.1 14.1 5747 452 1 6400 100 987.2 14.2 6494 500 10 7150 100 988.1 14.3 7184 550 10 7900 100 988.1 14.4 7903 595 10 8600 100 990.1 14.5 8549 Graph 2: A plot of the experimentally determined output voltage versus the theoretically determined, nominal, output voltage. Data for this graph is listed in Table 3. 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 V out vs. Nominal (Part 5) 0 2000 4000 6000 8000 10000 True Output Voltage: V out Series1 13
Gain Webster Part 6: Table 4: This table shows the values measured and calculated for the frequency decades. The data headings are the same as in Table 2 and are explained there. The nominal output voltage is dependent on V in (which varies marginally) and the resistances (which do not change). V in V in V out V out Freq. (Hz) Freq. (Hz) Gain Nominal Output 246 1 3480 10 99.4 0.005 14.15 3534.58 233 1 3280 10 299 0.05 14.11 3347.79 232 1 3280 10 500 0.05 14.14 3333.42 246 1 3560 10 987.2 0.05 14.47 3534.58 236 1 3360 10 2987 0.5 14.27 3390.90 234 1 3340 10 4995 0.5 14.27 3362.16 236 1 3320 10 9750 0.5 14.10 3390.90 244 1 3000 10 29170 5 12.30 3505.84 244 1 2320 10 48760 5 9.51 3505.84 251 1 1260 10 101000 50 5.03 3606.42 255 1 416 1 304500 50 1.63 3663.89 252 1 254 1 507600 50 1.01 3620.79 252 1 126 1 1008000 500 0.50 3620.79 232 1 82 1 1503000 500 0.35 3333.42 244 1 62 1 2000000 500 0.25 3505.84 Graph 3: A plot of gain versus the logarithmic frequency for the values listed in Table 4. 16.00 14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00 Gain vs. Log(Frequency) 0 1 2 3 4 5 6 7 Log(Frequency) Series1 14
Part 7: Image 3: This image of a square input signal at 1.002 khz was taken during the lab. It shows the output signal being a square wave as well. Image 4: This image of a square input signal at 100.0 khz was taken during the lab. It shows a change in the output signal wave form. 15
Image 5: This image of a square wave input signal at 1.008 MHz was taken during the lab. It shows the output as triangular wave. Image 6: This image of a triangular wave input signal at 1.000 khz was taken during the lab. It shows the output signal as having the same triangular waveform. 16
Image 7: This image is of a triangular wave at 99.80 khz. It shows the output signal as being sinusoidal. Image 8: This image is of a triangular wave at 1.008 MHz. It shows a sinusoidal output wave. 17
Part 8: Image 9: This image shows the square wave input signal at 1 khz frequency and 466 mv. From the image the rise time of the input signal can be determined to be 29.76 ns. Image 10: This image shows the output signal of the square wave at 1 khz frequency and 466 mv. The rise time of the output signal can be determined from this image to be 8.960 μs. 18
Analysis The gain or amplification with feedback can be calculated using equation 3 as follows: ( ) In Parts 2 & 3 the results shown in Table 1 were as expected. The values in the table are sensible, because of the two different resistances on inputs 1 and 2. When both of the inputs are connected with the same signal the output voltage should increase. To calculate the uncertainty associated with the gain we will use equation 6. The uncertainty in the input and output voltages are associated with the device, and are accepted to be 1 digit of the last readable place. ( ) ( ) ( ) ( ) The nominal output amplitude is calculated using equation 7. It is said to be nominal because it is the theoretical value for the output amplitude. The oscilloscope gives the true value. V A and V B are the input voltages and are equal for this case. ( ) ( ) ( ) The uncertainty in the nominal output amplitude is given by equation 8. The uncertainties on the resistors are given by the color coding on the resistors themselves. Since two resistors were used for R1, the uncertainty was estimated to be twice the value of the uncertainty of each individual resistor as determined from the resistance value recorded using a voltmeter. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ( ) ) ( ) ( ( ) ) ( ) ( ) ( ) ( ) ( ) 19
For the above equation, V A and V B are in units of volts, and resistances are in Ohms. In Part 5 the output voltage was changed to ±10V, to make the saturation value more pronounced. There was a systematic error in our experiment due to incorrect reading of the voltages. The values actually being recorded were the peak to peak values (on the oscilloscope) instead of trough to peak, resulting in values that are half of what needed to be recorded for saturation to take effect. The recorded values in the data tables were corrected for this error, but since the voltage was not at a high enough level saturation of the signal did not occur. The slew rate can be calculated using equation 9, where the V out is just the change in the output voltage cutting out the top and bottom 10% (just measuring the middle 80%). Specifically, in Image 10 the blue channel 2 line is the output, so starting at the first integer box (not measuring the half boxes on top and bottom) measuring upwards there are 4 boxes each with a value of 1.20 V. Therefore, V out = 4.80 V. ( ) ( ) Table 5: For both the input and output, the values for rise time and slew rate are listed. V in (V) V out (V) V out (V) Gain Rise Time (μs) Slew Rate (V/μs) Input Channel 0.466 3.5 0.1 7.5 0.02976 117.6075269 Output Channel 0.466 3.5 0.1 7.5 8.960 0.390625 In Graph 3, the change in gain can be seen versus the change in frequency. The gain is experimentally determined to be fairly steady (fluctuating slightly around 14) until a frequency of about 10 khz is applied. The gain begins to rapidly decrease as the frequency is raised even higher. This drop in gain is the effect of the amplifier being incapable of amplifying the signal at the frequency at which it is arriving. This effect is visible when using an oscilloscope, as shown in Images 3-10. The frequency value at which the signal can no longer be amplified can be determined by: ( ) 20
( ) ( ) This analytically determined value for the frequency at which the op amp can no longer amplify the signal is fairly consistent with the values found in Table 4. Some variation is to be expected due to equation 11 being only a rough estimate of the true frequency. 21
Discussion In Parts 2 & 3 the data is consistent with the values being within the range of acceptability. In Part 4 all of the data seems to be reliable. As the input voltage is increased, the output voltage jumps up for every 50 mv added to the input. The output voltage then successively increases by about 1000 mv for every 50 mv added to the input until the amount it increases by goes down to 100 mv in the last data point. The nominal output voltage seems to be close to the true recorded values. The uncertainties in the nominal don t quite make up the difference between the nominal and the true values. This could be due to slight voltage losses through the circuit that are, for the most part, negligible as the equation is based on an ideal amplifier. During Part 5, the saturation value was intended to be noticed by lowering the supply voltage, and going through the same steps as in part 4. Due to a systematic error, in which the voltages being applied were actually half of the intended values, this result did not occur. The saturation value for typical op amps is around one volt below the supply voltage 5, which is a reasonable estimate due to slight voltage loss across the circuit. In Part 6, Graph 3 shows that as the frequency increases the gain decreases. A gain of greater than one means that amplification is occurring, whereas a gain of less than one means that it is a passive circuit resulting in voltage loss instead of amplification. The data recorded in Table 4 is a testament to this. In Part 7, Images 3-8 show that as the frequency of the input wave is increased the output wave signal changes. Specifically for this op amp, a square wave input signal will begin to have a triangular wave output signal at around 100 khz. A triangular wave input signal will begin to have a sinusoidal wave output signal at around 100 khz. This is due to the slew rate of the op amp. The voltage can only be raised at a specific rate. Increased input frequency results in a stretched wave form that is even more pronounced at higher input frequencies. This is why it isn t noticeable at lower frequencies. At high frequencies the slew rate has a greater affect, which results in smaller gain, because the op amp cannot amplify the signal at the rate at which it is being received. The determined value for the slew rate (0.536 V/μs) is perfectly within the acceptable range. For the 741A op amp, the typical slew rate is 0.7 V/μs. The minimum is slew rate is listed to be 0.3 V/μs. 6 5 (Oliveira Sannibale, 2012) 6 (National Semiconductor Corporation, 2000) 22
Conclusion The data collected and calculations performed in the experiment give good insight into operational amplifiers and general circuitry. The first set of data shows the effects that grounding the different inputs has on the output voltage and gain. The second set of data shows the effects of solely increasing the input voltage, and allows us to calculate the theoretical (nominal) values for the output voltage. The third set of data was intended to show that changing the supply voltage affects the saturation value of the output, but no visible effect was present in the case of this experiment due to a systematic error. The fourth set of data shows that as the frequency is increased to a significant amount, the gain decreases, and the op amp no longer amplifies. The fifth set of data shows that the slew rate has a visible effect on the output wave signal. The final set of data shows the rise time from which the slew rate of the op amp was determined to be 0.536 V/μs. It can also be concluded that the slew rate has a relation to the frequency dependence of the gain. 23
Appendix A.1 Formula for the propagation of errors: ( ) ( ) ( ) ( ) 24
References Operational Amplifiers (Op Amps). (2001). Retrieved October 1, 2013, from siliconfareast.com: http://www.siliconfareast.com/opamps.htm Carter, B., & Brown, T. R. (2001, October). Handbook of Operational Amplifier Applications. Retrieved October 5, 2013, from Texas Instruments: http://www.physics.fsu.edu/courses/fall13/phy3802l/exp3802/mech_em/sboa092a.pdf National Semiconductor Corporation. (2000, August). LM741 Operational Amplifier. Retrieved October 6, 2013, from www.national.com: http://www.physics.fsu.edu/courses/fall13/phy3802l/exp3802/mech_em/lm741.pdf Oliveira Sannibale, V. d. (2012, December). Analog Electronics Basic Op-Amp Applications. Retrieved October 16, 2013, from California Institute of Technology: http://www.ligo.caltech.edu/~vsanni/ph5/pdf/ph5.chapter.basicopampcircuits.pdf PHY 3802L Experiments. (n.d.). Retrieved October 1, 2013, from FSU Physics: http://www.physics.fsu.edu/courses/fall13/phy3802l/exp3802/mech_em/opampexp1.pdf Research Solutions and Resources LLC. (2011, March 6). Control Amplifier Bandwidth and Slew Rate. Retrieved October 8, 2013, from Resources for Electrochemistry: http://www.consultrsr.com/resources/pstats/bwidth.htm 25