A Detailed Lesson on Operational Amplifiers - Negative Feedback

Transcription

1 07 SEE Mid tlantic Section Spring Conerence: Morgan State University, Baltimore, Maryland pr 7 Paper ID #0849 Detailed Lesson on Operational mpliiers - Negative Feedback Dr. Nashwa Nabil Elaraby, Pennsylvania State University, Harrisburg, The Capital College Dr. Elaraby is a aculty at Penn State Harrisburg since ugust 04. She received her PhD degree in Electrical and Computer Engineering rom Temple University in 04. She received her B.Sc and M.Sc. in Electrical Engineering rom lexandria University, Egypt. Her research interests include digital logic design using Field Programmable Gate rrays or massively parallel data computations, electronic circuit design, and neuronal data processing or Brain Machine Interace applications. c merican Society or Engineering Education, 07

2 Detailed Lesson on Operational mpliiers - Negative Feedback Nashwa Elaraby, Ph.D., Penn State Harrisburg bstract: Operational mpliiers present one o the important topics in electronic circuits courses. lthough they are widely taught, the model or the operational ampliiers with negative eedback coniguration is incomplete in most o the electronics textbooks 3 4. The closed loop gain or the non-inverting ampliier is given as /(+β), while the closed loop model or the inverting ampliier is usually not included. Instead nodal analysis is used to derive the expression or the inverting ampliier gain. Using dierent derivation paths and models causes conusion or the students, as the comparison is not provided rom the same point o view. The constant gain-bandwidth product is mentioned or the closed loop requency response or the ampliier, but it is not clearly stated that it only applies to the non-inverting ampliier coniguration. The missing details usually cause misconception or the students and mismatch between the lab results and their expectations. The main obective o this paper was to provide instructors with a detailed lesson on operational ampliiers using negative eedback, that can be applied in electronic circuit analysis courses or electrical engineering students and electrical engineering technology students. I. Introduction: Operational ampliiers with negative eedback have three modes o operation. The non-inverting ampliier, the inverting ampliier and the ltage ollower. The ltage ollower can be considered a special case o the non-inverting ampliier, but it will be considered separately in this paper, as it is a commonly used buer circuit that the students need to closely understand. The gain or the dierent ampliiers can be derived using the negative eedback coniguration. The main idea o the negative eedback is to reduce the potential dierence between the inverting and the non-inverting ampliier inputs to obtain a ltage at the output that is conined within the boundaries o the dc ltages biasing the internal transistors o the operational ampliier to operate in the orward active mode. The second section o the paper introduces the methodology used to complete the modeling details o the operational ampliier circuit with negative eedback. It covers the three negative eedback models and derives the expressions or the exact calculations o the closed loop gains in terms o the limited open-loop gain o the operational ampliier. The derivation o the ideal closed loop gains is veriied using the model. It also discusses the open-loop requency response o the operational ampliier. The methodology section also provides a transer unction equation or a compensated operational ampliier model, and shows how it is represented by a Bode plot equation. This step shows the students how to algebraically prove the roll o rate on the Bode plot. Lab procedures are included to test and veriy the requency response o a compensated operational ampliier. The closed loop requency response is then proved algebraically or both the inverting and the non-inverting conigurations, and lab procedures are included to test them. The third section provides the overall evaluation results o the applied lesson and the last section provides a discussion on the paper content. Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

3 I. Methodology: (a) Negative Feedback Models: The negative eedback coniguration aims at minimizing the potential dierence vd between the inverting and non-inverting inputs o the operational ampliier, and hence applying it as a linear ampliier with output values limited between the biasing ltages o the internal circuitry. With the negative eedback and an ideal open-loop gain o, we can assume that the dierential ltage vd is zero. For an ideal operational ampliier, the input impedance is ininity, which means that there is no current lowing into the inputs o the operational ampliier. The closed loop gain or the operational ampliier can hence be derived using simple nodal analysis as shown in igure. Virtually at vin vd 0 vin Virtually at vin ir ir Kircho 's Current Law : i 0 v R R in v o i R vin v R R R R R o v in ir Virtually at ground ir Kircho 's Current Law : i i vin 0 0 v R R R v o R R R R R o v in Fig. The negative eedback conigurations o an ideal operational ampliier: the ltage ollower, the noninverting operational ampliier and the inverting operational ampliier. The derivations o the closed loop gains assume an ininite open-loop gain and an ininite input impedance o the ideal operational ampliier. Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

4 For a non-ideal operational ampliier with inite open-loop gain the ollowing models in ig. can be applied to derive the closed loop gain. The addition operator signs are determined based on the connection to the inverting or non-inverting terminals. The negative eedback connection strives to cancel out the input signal contribution to minimize the dierential ltage vd. For the non-inverting ampliier the input is connected to the non-inverting input while the positive eedback signal is ed into the inverting input o the operational ampliier. For the inverting ampliier the eedback signal and a raction o the input signal are ed into the inverting input, but as the output is negative, the inverting input adds the absolute value o the eedback signal to the negative input signal. The α and β are ltage dividers ound by applying the superposition theorem or vin and respectively. where R R R where R R R R R R Fig. The negative eedback model o (a) the ltage ollower, (b) the non-inverting operational ampliier and (c) the inverting operational ampliier. Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

5 In the table, a thorough derivation o the closed loop gain expressions is presented. The ltage ollower The non-inverting ampliier The inverting ampliier vd vin vin vin vin v vd vin v vin vin vin vd vin vin vin vin v Table : The derivation o the closed loop gain o the dierent negative eedback conigurations using their models. To veriy the models used or the negative eedback conigurations the open-loop gain can be set to ininity or the ideal operational ampliier, and the derived closed loop expressions will be compared to the ones obtained using nodal analysis. In the derivation steps we neglect the term with respect to β assuming that with the ininite values o, the term β >>. The ltage ollower The non-inverting ampliier The inverting ampliier The ideal R R R R The closed loop gain based on the negative eedback model Veriying the closed loop gain model where R R R R R R R R where R R R R R R R R R R R R R R Table : Comparison o the negative eedback expressions obtained using the negative eedback model and the nodal analysis assuming ideal operational ampliier open-loop gain and input impedance. Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

6 (b) Open-Loop Frequency Response o a Compensated Operational mpliier: For this section the lab procedures or measuring the open-loop requency response and the unity gain bandwidth o the LM74 operational ampliier will be included. The main obective o this lab is to let students realize that the operational ampliier does not have a constant gain or all requencies, instead it is behaving like a low pass ilter with a dominant pole, governed by the ollowing equation: where 0 0 The low requency open - loop gain the dominant pole requency t lower requencies, it will be very hard to measure the gain, as the operational will reach saturation at the output or very low amplitudes o input ltages due to the high gain. s the requency increases, the gain decreases, and it will be possible to obtain an undistorted output ltage signal. The students will measure the gain and phase shit at dierent requencies between 0KHz and 500KHz. I any non-symmetrical clipping is observed on the output signal, positive or negative DC oset ltage in the range o a ew mv can be added to the input signal to cancel out the oset ltage causing the clipping. The range o requencies examined is much higher than the dominant pole requency (~0Hz), hence the magnitude and the phase equations can be approximated to the ollowing ormulas. It is important to measure the gain based on undistorted signals. 0 log db log Phase angle tan 90 This term sets the roll-o rate at -0dB/decade on the Bode plot applying logarithmic scale or the requency The students then plot () db versus the requency using a logarithmic scale or the requency. On the graph, the data points are extrapolated to ind the requency T at which the gain is unity or 0dB. To get the dominant pole requency, the graph is extrapolated to the requency at which the gain reaches the maximum value o the LM74 gain o 07dB. Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

7 The ollowing MTLB code can be used to plot the data points on the Bode plot and extrapolate it to obtain T and. Unity gain requency Dominant pole cut-o requency Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

8 (c) Closed-Loop Frequency Response: The operational ampliier with negative eedback also behaves as a low-pass ilter. To ind its bandwidth, the eedback models presented in section II will be applied. The closed-loop gain equations derived will be implemented by substituting or the open-loop gain by the unction (), and the comparing the equation to the general low-pass ilter transer unction. The ltage ollower: 0 0 where (0) B.W. 0 ltage ollower 0 0 The non-inverting mpliier: 0 0 where (0) B.W. 0 non-inverting 0 For the ltage ollower and the non-inverting ampliier the Gain-Bandwidth product is a constant value: 0 0 B.W. 0 B.W db db B.W. - 0 log Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

9 The inverting mpliier: where (0) B.W. 0 non-inverting 0 0 For the ltage ollower and the non-inverting ampliier the Gain-Bandwidth product is a constant value: 0 B.W. 0 The gain-bandwidth product is not a constant value. It depends on the orward attenuation α, which varies or every closed loop gain. Lab procedures: The lab procedures or the closed loop requency gain are divided into two main sections: the closed loop gain or the non-inverting ampliier and the closed loop gain or the inverting ampliier. Each section is subdivided into two parts, one showing that the operational ampliier with the negative eedback is behaving as a low-pass ilter, and then calculating the GBW product or dierent gain values. In the discussion section o the lab report, students compare the requency responses o the two coniguration o negative eedback. They compare the GBW products. They also compare their requency responses to that o the open-loop coniguration. They plot the closed-loop low requency gain versus the requency on a logarithmic scale and explain the roll-o rate using the constant GBW product relation or the non-inverting ampliier. They show that this relation does not hold or the inverting ampliier. The set o labs explaining the dierent negative eedback models and the requency response, help students build a concrete oundation or the understanding o the operational ampliier application as a linear ampliier in the dierent modes. Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

10 Part : Closed-loop Frequency Response o a Non-Inverting mpliier: () Connect the circuit shown in igure. Veriy the gain as the ratio o Vout/Vin using a low requency sinusoidal signal o an appropriate amplitude. (aid clipping, but remember that the larger the signal the more accurate your recordings will be) () What is the eedback attenuation β? (3) Measure the gain at dierent requencies between 00Hz and MHz, and plot the closedloop requency response. Plot the graph (Take at least 0 readings). Measure the bandwidth by gradually raising the requency until the gain drops to times its nominal value (The nominal value (0) is the value o the gain at very low requencies). The cut-o requency needs to be one o you recordings. (4) For any non-inverting ampliier circuit, the gain o the ampliier is CL. Sketch the model or the non-inverting ampliier, and prove the equation or the closedloop gain. t high values o the closed loop gain can be approximated to be /β. Prove that, and show how it applies to your circuit results. (5) Using the transer unction o the operational ampliier Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU (0) ( )

11 prove that the bandwidth or the non-inverting ampliier is (0) c. (6) Prove that the Gain-Bandwidth Product (GBW) or the non-inverting ampliier is a constant value equal to GBW (0).The GBW value also represents the Unity gain Band Width T, which is the bandwidth at a closed loop gain o (Voltage ollower). (7) Use dierent values o R to obtain closed-loop gains rom 0dB to 50dB with a step size o 5dB. For each value o the low-requency closed-loop gain, chose an appropriate eedback resistor value, and veriy gain by calculating the ratio o Vout/Vin., using a low requency sinusoidal signal o an appropriate amplitude (aid clipping at the output) For each gain value, measure the bandwidth by gradually raising the requency until the gain drops to times its nominal value. (8) Use your measurements to create a graph o the gain (in db) on the y-axis versus the bandwidth (logarithmically scaled) in Hz on the x-axis. PD c (0) : (0) c 0 log 0 log : (0) 0 log c c (0) 0 log db db (9) On your graph, extrapolate your data points to the location where gain would be unity (zero db). t this point, the unity gain bandwidth will be equal to the value o the GBW. Then extrapolate your graph to the point where the gain would be 0,000 (07dB). t that point, the bandwidth should be that o the 74 in an open-loop coniguration. (0) Connect a ltage ollower and increase the requency until the gain becomes to ind the unity gain BW T. What might be your limitation or the gain measurement at the high requency? Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

12 Part : Closed-loop Frequency Response o an Inverting mpliier: () Connect the circuit shown in igure. Veriy the gain as the ratio o Vout/Vin using a low requency sinusoidal signal o an appropriate amplitude. (aid clipping, while the larger the signal the more accurate will be your recordings) () What is the eedback attenuation β and the multiplier α? (3) Measure the gain at dierent requencies between 00Hz and MHz, and plot the closedloop requency response. Plot the graph (Take at least 0 readings). Measure the bandwidth by gradually raising the requency until the gain drops to times its nominal value (The nominal value is the value o the gain at very low requency). The cut-o requency needs to be one o your recordings. (4) For an inverting ampliier circuit, the gain o the ampliier is CL. Sketch the model or the non-inverting ampliier, and prove the equation or the closedloop gain. t high values o the closed loop gain can be approximated to be -α/β. Prove that, and show how it applies to your circuit results. (5) Using the transer unction o the operational ampliier (0) ( ) prove that the bandwidth or the non-inverting ampliier is (0) c. PD Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

13 (6) Prove that the Gain-Bandwidth Product (GBW) or the inverting ampliier is equal to GBW (0). (7) Use dierent values o R to obtain closed-loop gains rom 0dB to 50dB with a step size o 5dB. For each value o the low-requency closed-loop gain, chose an appropriate eedback resistor value, and veriy gain by calculating the ratio o Vout/Vin., using a low requency sinusoidal signal o an appropriate amplitude (aid clipping at the output) For each gain value, measure the bandwidth by gradually raising the requency until the gain drops to times its nominal value. (8) Use your measurements to create a graph o the gain (in db) on the y-axis versus the bandwidth (logarithmically scaled) in Hz on the x-axis. c : (0) : 0 log 0 log : 0 log 0 log c (0) (0) 0 log c (0) 0 log PD (9) The GBW or the non-inverting ampliier is a constant value. What about the inverting ampliier? Explain. III. Results: I have taught circuits courses covering the operational ampliier application as linear ampliier or both Electrical Engineering and Electrical Engineering Technology student or at least three years both in the spring and all semesters. With the detailed modeling o the operational ampliier circuits and emphasizing the dierence between the inverting and non-inverting conigurations, I had much better results or the exams and quizzes. The lab experiments go aster and are successully completed with more than 95% o success rate compared to 60% in the past when we relied on the incomplete theoretical explanation and models o the textbooks. I have asked students whether it was easier to study rom the textbook or rom the class notes, and they answered that the detailed comparison o the dierent negative eedback conigurations provided in the class notes, made it easier or them to remember their dierences and similarities. Beore adding the detailed lesson to my class notes, many students were applying the constant GBW product or both the inverting and the non-inverting ampliier conigurations, c Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU

14 and they were rustrated when the lab results were not aligned with their calculations. With the detailed explanation the rate o these instances has signiicantly decreased. IV. Conclusion: Modeling the negative eedback conigurations o operational ampliiers and the requency response o the ampliier circuits is a topic that needs clariication in most o the textbooks. detailed comparison o the three conigurations: Non-inverting ampliier, inverting ampliier and ltage ollowers are needed to give the students a concrete theoretical basis to understand the circuit behavior. The main obective o this paper was to provide instructors with a detailed lesson on operational ampliiers, that can be applied in electronic circuit analysis courses or electrical engineering students and electrical engineering technology students. It presents a thorough lesson with lab procedures on the negative eedback models as well as their requency responses, which has improved student's understanding o the operational ampliier behavior and modeling. Reerences:. Donald. Neaman, Microelectronics: Circuit nalysis and Design, 4 th Edition McGraw-Hill 00, ISBN Basic Operational mpliiers and Linear Integrated Circuits, nd edition, Floyd and Buchla, Prentice Hall, del S. Sedra and Kenneth C. Smith Microelectronic Circuits, Oxord Series in Electrical & Computer Engineering, 6th Edition 4. Richard C. Jaeger, Travis N Blalock, Microelectronic Circuit Design, 4th edition, McGraw-Hill 0, ISBN Spring 07 Mid-tlantic SEE Conerence, pril 7-8, 07 MSU