Operational Amplifiers - Assignment Example

Operational Amplifiers Contents Page No Introduction Objectives Theory Calculations Circuit 1 Calculations Circuit 2 Procedure Results Interpretation of results Comparisons Conclusions Bibliography Operational Amplifiers Abstract Bode analysis was used to investigate the difference between theoretical and practical results of frequency gain and phase gain of operational amplifiers. The results show that there is slight variation in the theoretical findings and the real findings. This is because in circuit 2, (open loop) the theoretical bode plot shows an infinite gain where as in practical situation the gain is not infinite as there is continued variation until the gain becomes zero. It was also found that negative feedback stabilises signal gain. Introduction An operational amplifier that is ideal has several characteristics that are unique. In case the device is to be used as a gain block, an amplifier that is ideal should have infinite gain. Similarly, the input impedance of the amplifier should also be infinite so as not to draw any power from the driving source. The most commonly used method for analysing stability of an amplifier is the Bode analysis. The basis of measurement here is creation of an open loop magnitude and phase plot to attain the stability for a closed loop. These are also indicators of gain and phase margin. Derivation of the phase margin is done by finding the intersection of the unity gain frequency response of closed loop curve to the open loop response curve (Schmid, 1995). At this frequency the phase will be read from the phase plot. Then the value gotten is subtracted from one hundred and eighty degrees to get the desired phase margin. Gain margin can also be determined in the magnitude plot by the frequency at 180 degrees. Objectives 1. The aim of this experiment is to use Bode analysis and practical measurement to investigate the frequency response (gain and phase) of the two operational amplifier circuits shown below. 2. Use Multisim software package to test both circuits shown in Figures 1 and 2 should be 3. Construct the circuits of Figures 1 and 2 on the ‘bread board’ and test them as REAL circuits. 4. Compare the results for the REAL circuit with its virtual response. Figure 1 Figure 2 Theory Operational amplifiers are linear devices possessing all the qualities that are required to have ideal amplification of direct current. Operational amplifiers are used for filtering or conditioning signal. They can also used to subtract, add, integrate and differentiate mathematical operations. An ideal amplifier consists of three terminals. The negative terminal is the inverting input while the positive terminal is the Non-inverting input. The last terminal is the output port of an operational amplifier. Characteristics of an ideal or theoretical amplifier 1. The open loop gain is infinite. The open loop is the gain of an operational amplifier without negative or positive feedback. 2. The input impedance is infinite. This prevents any current from flowing from the supply source into the input circuitry of the amplifier. 3. The circuit has zero output impedance to facilitate maximum supply of current to the load. 4. There is an infinite frequency response. This means that it can amplify any magnitude of frequency. How gain is derived in a non-inverting amplifier circuit as shown in Figure 3. Figure 1 The values of the resistors R1 and R2 determine the amount of output fed back to the amplifier in the above circuit. .................................................. 1 Then, β Where β feedback factor Therefore, .................................................................................................... 2 The open loop gain for an operational amplifier (AVOL) is given as: …………………………………………………………………….. 3 From Figure 3, it can be seen that, ……………………………………………………………………………………………………… 4 Making VO the subject of the formula in equation 3 we get, But, Therefore, .......................................................................................... 5 But, from equation 2, Therefore, Expanding the equation above we get, Factorizing the equation, ............................................................................................ 6 For closed loop gain (AVCL): ..............................................................................................7 Dividing the numerator and the denominator by AVOL we get, As the open loop gain for an operational amplifier is infinite, and then the reciprocal of AVOL is 0, we get, .............................................................................................................. 8 Hence: But, When the same circuit is considered in terms of impedance, R1 = Z1, R2 = Z2. Capacitors and inductors effectively change their impedance dependent upon frequency (ZC = 1/jωC). With capacitors in the circuit as in Figures 1 and 2, β will be determined by frequency. Calculations Circuit 1 Figure 6 Circuit gain; Break frequencies: Therefore, Low break frequency, And High break frequency, These results would produce the Bode plot shown below in figure 7. Figure 7 Circuit 2 Figure 4 Circuit gain: Break frequencies occur where the real term = the imaginary term: High break frequency, ∴ Low break frequency, ∴ These results would produce the Bode plot shown below in figure 5. Figure 5 Procedure 1. Simulate the circuits in turn using Multisim as specified in the objectives. The Operational Amplifier used was a 741. 2. Use the following instruments as set up in the figure below to carry out the experiment InstekGOS-6112 Oscilloscope Signal Generator Fluke 89 Multimeter Power supply Figure 8. 3. Select a suitable input voltage was and set the Frequency of the signal generator to 500Hz, 1 kHz, 5 kHz and 10 kHz in turn. 4. Record the results in the table format as shown in tables 1 and 2 below. 5. Analyse the output voltage and phase shift of the two circuits using the oscilloscope which is used to produce two Bode plots showing; the magnitude of the gain against frequency the phase response against frequency 6. Compare the results of the REAL measurements with the Bode plot Analysis and the calculated theoretic (ideal) values. 7. Print the Bode plots from the oscilloscope. 8. Construct the circuits using the Bread Board and the available components. Results Circuit 1 Frequency (Hz) VIN (v) VOUT (v) Gain Gain (dB) Phase Angle (0) 500 0.404 3.20 7.92 17.98 Lagging 30 1 k 0.409 2.35 5.75 15.19 Lagging 37 5 k 0.414 0.90 2.17 6.72 Lagging 28 10 k 0.419 0.78 1.88 5.48 Lagging 16 Table 2 Circuit 2 Frequency (Hz) VIN (v) VOUT (v) Gain Gain (dB) Phase Angle (0) 500 0.4 6.8 17 25 Leading 50 1 k 0.4 12 30 30 Leading 70 5 k 0.2 17.5 87.5 39 Leading 7 10k 0.2 14.4 72 37 Lagging 27 Table Interpretation of the results Circuit 1 The gain of the amplifier is constant as the frequency of the signal increases from 1 Hz to around 200Hz. As the frequency of the signal increases beyond the low break frequency i.e. 200Hz the gain increases until it reaches its maximum. The frequency at which the amplifier gain reaches maximum value is known as resonant frequency or of the high break frequency. Beyond this high break frequency the amplifier gain steadily declines to zero. Therefore it is clear from these observations that the frequency response is poor i.e. gain is constant over a small range of frequency. It then follows that the amplification of the signal will be disproportionate. Hence there will be distortion of frequency. Circuit 2 The gain of the amplifier is constant as the frequency of the signal increases from 1 Hz to around 200Hz. As the frequency of the signal increases beyond the low break frequency i.e. 200Hz the gain decreases until the frequency of the signal reaches the high break frequency. Beyond this high break frequency the amplifier gain again remains constant until a certain frequency when it declines to zero. Therefore it is clear from these observations that the frequency response is good i.e. gain is constant over a wide range of frequency. It then follows that the amplification of the signal will be proportionate. Hence there will be no distortion of frequency. Comparison of results of REAL circuit with its virtual response Similarities in circuit 1 1. In both virtual and real circuits the gain does not start at zero. This gain is initially almost constant as the frequency is increased up to a certain break frequency. The low break frequency of the theoretical value is 332 Hz while that of the real results ranges between 200Hz to 1000Hz. Then there is a steady rise in gain after the low break frequencies up to the high break frequencies. The corresponding high break frequencies of the theoretical gain are 3.29 kHz while those of the real results are in the range of 1.8 kHz to 11 kHz. Then the values of gain steadily fall as the frequency increases beyond the high break frequency up to zero. 2. In both virtual and real circuits, the phase plots start at 00. Then as frequency is increased there is a steady rise in the phase angle up to a certain maximum angle. From this maximum angle there is a decline in phase angle as the frequency increases until the phase angle is zero. Then phase angle continues to fall below zero as the frequency increases up to a certain frequency when the phase angle will again start to stabilize and remain constant. Differences in circuit 1 1. In the theoretical values, after the high break frequency there is no further decline in the gain that is the system remains stable. On the other hand, in real results it is clear that after the high break frequency the value of the gain steadily declines until it reaches zero to meet the frequency axis at around 1,1000,000 Hz. This means that the system does not attain stability. Therefore, the theoretical values suggest a stable amplifier where there is no distortion of the signal but in reality the signal is distorted. 2. There is generally variation in the values of break frequencies between the real and theoretical plots. This is explained by the nature of both curves. The theoretical plot has sharp breaking points where as the real plots have smooth breaking points. That is the real plot curves are smooth while the theoretical ones are rough. Similarities in circuit 2 1. In both virtual and real circuits the gain does not start at zero. This gain is initially almost constant as the frequency is increased up to a certain break frequency. The low break frequency of the theoretical value is 803.8 while that of the real results ranges between 800Hz to 1000Hz. Then there is a steady decline in gain after the low break frequencies up to the high break frequencies. The corresponding high break frequencies of the theoretical gain are 4.23 kHz while those of the real results are in the range of 1.9 kHz to 11 kHz. Then the values of gain remain constant in both cases. 2. In both virtual and real circuits the phase plots start at 00. Then as frequency is increased there is a steady decline in the phase angle up to a certain minimum angle. From this minimum angle there is an increase in phase angle as the frequency increases until a certain frequency when the phase angle will again start to decline forming a generally sinusoidal curve on the plot. Differences in circuit 2 1. In the theoretical values, once the gain is constant after the high break frequency there is no further decline in the gain. On the other hand, in real results it is clear that after the high break frequency the value of the gain remains constant up to around 500,000 Hz then the gain declines further to intercept the frequency at around 1,1000,000 Hz. 2. It is also clear from the phase plot that under theoretical values the sinusoidal curve formed is almost symmetrical but in the phase plots of the real values the sinusoidal curve is asymmetrical. 3. There is general variation in the values of break frequencies between the real and theoretical plots. This is explained by the nature of both curves. The theoretical plot has sharp breaking points where as the real plots have smooth breaking points. That is the real plot curves are smooth while the theoretical ones are rough. Conclusions From this experiment, we learn that voltage gain of an amplifier varies with frequency. This is because of the reactance of the capacitors in the amplifier circuit changes with frequency of the signal thereby affecting the output voltage. The difference between the high frequency and low frequency is known as bandwidth. These frequencies can also be termed as limiting frequencies. When there is no distortion in amplification then the frequency of the signal must lie within these frequencies. The performance of an amplifier depends on the considerable extent upon its frequency response. While designing an amplifier it is important to take the right steps so as to ensure that gain is constant upon a specified range of frequency. It is practically important to assign amplifier gain a unit. This unit used to express amplifier gain is called decibel. It is denoted as db. A bel power gain is the logarithm to base 10 of power gain. There are several advantages of expressing gain in decibel. One of the advantages is that the decibel is a logarithmic unit. This coincides with the fact that our ear response is also logarithmic. This means that the loudness of sound heard by the ear is not determined by the intensity of sound but by the logarithm of the intensity of sound. Thus if the intensity of sound given by the speaker (i.e. power) is increased 1000 times, our ears hear a tripling effect (log10 1000 = 3). This will be as if loudness was tripled instead of made 1000 times. Hence this system tallies with the natural response of our ears. When negative feedback is applied, the actual voltage applied to the amplifier is very small as can be observed from both tables 1. Negative feedback circuits have got several advantages. One of such advantages is that the gain is stable. This can be seen from the equation below; The above equation means that in order for the negative feedback in the amplifier to be effective, then the value of >>1 so that the equation above can become, This means that gain will depend on feedback factor that is the characteristics of the feedback circuit. As the feedback circuit is naturally a potential divider (a network of resistors), the amplifier gain becomes independent of the ageing of transistor, temperature, variation in voltage, and frequency. Another advantage of negative feedback is that it minimizes non-linear distortion in large signal amplifiers as can be seen in the gain plot for circuit 1. This can be proved mathematically as; Where; D is the distortion in amplifier with negative feedback is the distortion in amplifier without feedback This therefore means that when negative feedback is introduced on an amplifier then distortion is reduced by a factor. Negative feedback also improves frequency response (Sedra, 1998). This can be drawn from which shows that voltage gain of the amplifier is independent of the frequency of the signal. According to Sedra the result of this is that voltage gain of the amplifier will be constant over wide range of signal frequency. The frequency response of the amplifier is therefore improved by negative feedback. Finally, negative feedback modifies output and input impedances. The input impedance is increased by negative feedback where as output impedance is decreased by negative feedback of the amplifier. This is important since it enables the amplifier to carry out an additional function of matching impedance. This can be shown mathematically as; Where; is the impedance without feedback. is the impedance with negative feedback Practical operational amplifiers such as the one used in this case do not have a bandwidth or an infinite gain. They have typical open loop gain. Bibliography Sedra / Smith, 1998. Microelectronic Circuits, Fourth Edition. Oxford: Oxford University Press. Schmid, R, “Technique targets board parasites”, EDN April14, 1994 page 147. Read More