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08/04/2014 PHAS1241 Closed Loop Gain of an Operational Amplifier Nisha Lad Abstract Operational Amplifiers (Op-Amp) are incredibly useful integrated circuits which are used for signal processing. The gain in amplification of a signal by an Op-Amp is highly important in electromagnetics, such as in loudspeakers and antennas. An experiment was conducted to investigate the effects of how the closed loop gain of an Op-Amp circuit varies with frequency for different feedback values. It was found that at high values of feedback, the bandwidth was larger compared with a low feedback value where the equivalent open loop gain was obtained. I. Introduction Electronic circuits can be classified into two types: analogue and digital. Digital circuits are used in computers where two signal levels are needed, whereas in analogue circuits, such as the Op-Amp, electrical signals can vary continuously. IA. Where the closed loop differential input signal is [V in - Vout] [1]. In this case the desirable effect is achieved, as G(f) decreases with increasing frequency which in turn increases the differential input signal [Vin - Vout] such that Vout, and hence the gain remain constant over a large range of frequencies, known as the bandwidth. Theory The purpose of an Op-Amp is to amplify a weak signal resulting from the difference in voltage between two inputs, known as the non-inverting (Vn) and inverting (Vi) inputs, giving rise to its name ‘differential amplifier’. The Op-Amp requires power to amplify the small input-signal (Vin) and hence the maximum possible output-signal (Vout) obtained, as a function of frequency, is described by Equation 1. Where G(f) is the equivalent ‘open-loop’ gain when there is no feedback and [Vn-Vi] is the differential voltage [1]. In practice the ‘open-loop’ mode is not used due to the fact excessively high gains are obtained at low frequencies across a short bandwidth. The desirable effect is generally a smaller gain that is constant over a wide range of frequencies. This is achieved by ‘feeding back’ a fraction of Vout into the inverting input of the Op-Amp, shown in Figure 1, so that its input depends upon its output and thus results in the formation of a closed loop circuit. The ratio between the input potential and output potential is described by equation 2 and equation 3 describes the relationship between Vout and the gain if a fraction of Vout, , is fed into the inverting input Vi, shown in Figure 1. Fig. 1 Circuit diagram of the non-inverting amplifier (triangle) used in the investigation. The circuit consists of a potential divider resistor network used to feedback a fraction, , of Vout into the inverting input (Vi) resulting in differential amplification. [1] Manipulating (3) gives the closed loop gain described by equation 4. Consequently, at lower frequencies where G(f) is much greater than unity, the gain is approximated by equation 5. 1 08/04/2014 PHAS1241 Where Glf indicates the low frequency gain of the closed loop circuit and is independent of frequency [1]. The closed loop gain, Glf, will remain constant with increasing frequency so long as G(f)>>Glf. This relationship breaks down when the open loop gain, G(f), reduces until G(f)Glf. Through manipulation of the potential divider relation given by equation 6, the low frequency gain is therefore determined via equation 7. II. Prior to starting the experiment, calibration of the Cathode Ray Oscilloscope (CRO) was performed for both Channels 1 (input) and 2 (output) in order to ensure inaccuracy in readings, and hence any systematic errors resulting from skewed measurements, were not introduced. It was ensured that the variation knobs were not rotated after calibration, as this would also introduce systematic errors in measurements, and that the oscilloscope inputs were set to measure DC signals to compensate for any DC offset. Experimental Method An experiment was conducted to investigate how the closed loop gain of the Op-Amp circuit varies with frequency for the low frequency gain, Glf, values of ≃10, 100 and 1000. This was achieved by changing the configuration of resistors R1 and R2 (shown in Figure 2) and using (7), hence the fraction of Vout, , which was fed back into the inverting input (Vi) was varied, whilst a small oscillating signal was fed into the non-inverting input (Vn). The Vout and Vin were then measured over a large range of frequencies and the corresponding gains were calculated. The circuit used in this investigation is shown in Figure 2 and the Op-Amp in Figure 3. Fig. 3 Schematic of the 8-pin 741 Op Amp integrated circuit as an electrical component used in the investigation. The triangle represents the amplifier. Pins 4 and 7 correspond to the connection of the power supply, 2 and 3 correspond to the feedback resistor network and 6 to the oscilloscope [2]. Initially, the circuit was connected to set up a feedback loop as shown in Figure 2, with a resistor configuration of R1=1MΩ and R2=10kΩ, resulting in Glf ≃ 100 by using (7). The oscilloscope was adjusted to read an appropriate sine wave signal for both Vin and Vout. The frequency was varied from the signal generator from 10Hz to 1MHz and measurements were taken from channels 1 and 2 to obtain Vin and Vout respectively. Subsequently, the gains were calculated using (2), thus a logarithmic graph was plotted of log(gain) as a function of log(f(Hz)), as shown in Figure 4. In order to reduce the proportion of random uncertainties, signals displayed on the oscilloscope were adjusted to be as sharp and as faint as possible. Furthermore, channels 1 and 2 were both triggered in order to ensure repeated signals displayed were in phase for ease of measurement. This procedure was repeated for resistor configurations of R1=1MΩ and R2=1kΩ, resulting in a Glf ≃ 1000 and finally Fig. 2 Schematic diagram of the 741 Operational Amplifier circuit board and resistor network connected to a synthesised function generator (ISO-Tech 1Hz-1MHz) and oscilloscope (ISO-Tech ISR622 20MHz) used in the investigation. Yellow arrows represent an example configuration of resistors R1 and R2 via yellow leads to allow a fraction to be fed back [2]. 2 08/04/2014 PHAS1241 R1=100kΩ and R2=10kΩ, resulting in a Glf ≃ 10. This enabled a different fraction of Vout, , to be fed back and the resulting gains were calculated. III. remained constant over a much greater bandwidth, as seen in Figure 4. The input voltage Vin remained constant at (10.0±0.5) mV over the three resistor configurations, as expected. Results and Analysis It was found that for a larger feedback, and hence a smaller Glf value, the larger the bandwidth over which the gain remained constant. After a certain threshold value, which depended on the value of feedback, the gain decreased for higher frequencies. This is shown in Figure 4. Additionally, it was also found that the results obtained coincided with the theory, in which the smaller feedback value, and hence larger the Glf, very high gains were obtained which remained constant over a short bandwidth compared to larger bandwidths for lower Glf’s. For example, for Glf ≃ 1000, the equivalent open-loop gain was obtained in which a large gain remained constant over a short bandwidth from 10-500Hz at a value of 950.00±68.97, which then decreased with increasing frequency, shown in Table 1. The uncertainty in the synthesized function generator was 40 parts per million [3], hence the uncertainties in f(Hz) were negligible compared with the values of f(Hz), for this reason they are not shown in Tables 1,2 and 3. f(Hz) Vout (±0.05V) Gain 10 1.00 100.00±7.07 5000 1.00 100.00±7.07 7500 0.84 84.00±6.53 Table. 2 Results obtained from a feedback of Glf ≃ 100. The gain obtained remained constant from 10-5000Hz and then began to decrease with increasing frequency up to 1MHz [2]. f(Hz) Vout (±0.01V) Gain 10 0.11 11.00±0.74 75000 0.11 11.00±0.74 100000 0.10 10.00±0.51 Table. 3 Results obtained from a feedback of Glf ≃ 10. The gain obtained remained constant from 10Hz to 100kHz and then began to decrease with increasing frequency up to 1MHz [2]. f(Hz) Vout (±0.50V) Gain 10 9.50 950.00±68.90 500 9.50 950.00±68.90 1000 5.00 500.00±55.90 Table. 1 Results obtained from a feedback of Glf ≃ 1000, where distortion in signals became prominent; hence the uncertainties were larger than propagated values. The gain obtained remained constant from 10-500Hz and then began to decrease with increasing frequency up to 1MHz [2]. Fig. 4 Plot of log(gain) as a function of log(Frequency(Hz)) from the data obtained for low frequency gain values of G lf ≃10, 100 and 1000. [2]. The results acquired from Glf ≃ 1000 were similar for the resistor configuration for Glf ≃ 100, where the gain obtained remained constant over a larger bandwidth of 10-5000Hz at a value of 100.00±7.07. It then decreased with increasing frequency shown in Table 2. A similar behaviour was also obtained for Glf ≃ 10, where the gain remained constant at a much lower value of 11.00±0.74, although the bandwidth was much greater for frequencies of 10Hz-100kHz. This data obtained is shown in Table 3. Therefore, it was observed that for each decreasing Glf, the gain obtained was much smaller than the previous and From Figure 4, it was found that the rate at which the gain decreased were not the same for all Glf values. This did not support the theory given by (2), and may be a result of signals being susceptible to noise at much greater frequencies (≃ 105Hz) as the Op-Amp becomes unstable. Consequently, uncertainties were adjusted to be greater than those propagated at higher frequencies to compensate for signals becoming more 3 08/04/2014 PHAS1241 distorted, due to the fact difficulty in taking precise measurements became apparent and obtaining a reliable noise-free output was harder than predicted. Subsequently, suitable improvements include taking repeat readings, which would improve the reliability of the investigation and reduce the proportion of random errors in results. Additionally, taking more measurements at smaller intervals of frequency would make the curves more defined, particularly about the turning point. This would also increase the accuracy of the investigation and make anomalous data points easily detectable. In order to reduce the amount of noise obtained, hence acquire less distortion and precise measurements at greater frequencies, a power supply with a greater voltage could be used as this would produce higher amplitude signals. IV. V. References [1] Dr P. Bartlett, “Experiment NX 741, Operational Amplifiers”, 13/01/2014, (UCL Department of Physics and Astronomy, Teaching Laboratory 1) [2] Nisha Lad, Lab Book 2, Experiment 1, “Operational Amplifiers” (UCL Department of Physics and Astronomy, Teaching Laboratory 1) [3] “Synthesized Function Generator User Manual, 82RS-21200M01”, pp 11, (UCL Department of Physics and Astronomy, Teaching Laboratory 1) Conclusions From the results obtained for each Glf, the gain, G, was found to be constant over a certain range of frequencies, known as the bandwidth, which then steadily decreased for increasing frequencies and eventually reaching zero. According to Figure 4, the bandwidth is shorter for a higher Glf and is larger for a low Glf. This is in agreement with the theory due to the fact that the amplitude gain would be at its maximum so long as G(f) >> Glf where G(f) is the equivalent openloop gain when is no feedback. Hence, when Glf is very large (1000), there is a small bandwidth over which G(f) remains constant and conversely if Glf is small (10), there is a much greater bandwidth at which G(f) remains constant. The results obtained are in accordance with what was expected, excluding the noise, and were comparable with colleagues’ results. Possible further work includes investigating other resistor configurations, in determining whether they follow the behaviour as obtained in Figure 4. Additionally, the Op-Amp can be investigated further with other electrical components, such as how it can be used to amplify the potential difference between two LDRs, which in turn allows current to flow to a motor. 4