1. Introduction
The open-circuit-time-constant (OCTC) method is an easily applied technique to estimate an approximation of an amplifier’s bandwidth. The importance of the method lies not only in the fact that it gives a low-error approximation of the
first Bode diagram (dominant) pole-frequency by just applying a relatively simple algorithm but also in its capability to provide appropriate information on the major causes that limit the amplifier’s bandwidth. This is such an important advantage that overtakes the usefulness of frequency response (ac) simulation, since the clues OCTC provides can be used as an effective tool for a large variety of amplifier types [
1,
2,
3]. This article is an extended version of the paper already published in MOCAST 2019 proceedings.
The significance of the OCTC method briefly described above leads us to utilize it as one of the most essential chapters in educational electronics. The undergraduate electrical and electronic engineering students have to deeply understand the OCTC algorithm process and take advantage of it at upper-level courses, when they will be asked to design their first amplifiers with specific properties. This will contribute to their experience as electronic engineers and their post-graduation professional or academic course.
In order to be able to deeply understand and properly apply OCTC when asked (circuit analysis) or needed (circuit design), the undergraduate students need to have a strong background in electronics. This a-priori knowledge and experience includes basic circuits and systems analysis (transient and steady-state node/loop method application), mathematical formulas (common differential equations and Laplace transform), BJT and MOSFET transistor DC and AC properties and basic DC supply and identification and analysis of amplification stages. In the Department of Electrical and Computer Engineering of National Technical University of Athens (NTUA), those subjects are covered by the “Introduction to Circuit Analysis”, “Introductory Electronics and Telecommunications Lab”, “Electronics I", “Electronics II” and “Circuit and Systems Theory” courses, being taught during the three first years of the bachelor degree program. The OCTC method is taught at the “Electronics III” course during the fourth bachelor degree year (7th semester) and it is expected that the students already have basic experience in the electronic analysis field. We choose this specific subject because OCTC is a basic part of the course “Electronics III” and in addition is a great example of a design tool for electronics. The OCTC method as a tool for circuit analysis and amplifier design enables students to test and repeat basic electronics knowledge while at the same time understanding the utility of approximate evaluations of electronic circuit behavior. It is also a topic offered for comparison between simulation and analytical-approximation solution.
As university educators, we have been teaching “Electronics III” as an optional course, enrolled by students that wish to follow a discrete electronics or analog IC designer path, for the last five years. In the last three years, the course grading policy has been changed by adding an LTspice simulation lab, in order to assist the students to better understand how the theoretical methods being taught are linked to some basic applications as well as taking their first steps as electronic designers.
The OCTC method is a mandatory part of this educational process. However, teaching experience has shown that the students have a characteristic difficulty in understanding and applying it even when they are asked to in simple quizzes. The main causes do not appear to stem from a misunderstanding of the algorithm process itself as someone would expect. Instead, we have revealed that an important percentage of them actually apply OCTC whilst lacking fundamental amplification stage analysis knowledge that could really simplify the overall procedure. It was observed that when they apply OCTC, they prefer writing complex equation systems to compute each parasitic capacitor’s impedance, leading them to mistakes. Exam problems or exercises of finding equivalent resistance are solved by many students in an unnecessarily complicated way. This is because they have not embodied or learned to use known step-by-step properties as key tools. On the other hand, there is often a question of whether the students understand the usefulness of the OCTC method as a design tool, because they see it as a method to be applied only when asked for as part of an exercise.
The purpose of teaching this module is to alleviate the above phenomena. This can be done using teaching techniques such as frequent reminders of the basic tools, encouraging the use of amplifier properties, explaining design problems and using simulations. It has been observed that the students tend to address electronic circuit problems in a rather local and sequential way with no reference to the appropriate principles, rules, and methods. A frequent reminder of the basic tools taught in previous semesters is important.
It is helpful to encourage the use of amplifier properties through examples, the first step by step and then as a complete application. Memorization of specific circuits, gain formulas, and key results may play a crucial role in students’ ability to successfully solve typical amplifier circuits. The findings from the teaching feedback suggested that many students likely did not possess a robust understanding of the behavior of different basic amplifier circuits, even after instruction on basic circuits has been completed (in previous semesters). For this reason, we were interested in developing a task in which students would be forced to think deeply about the currents and voltages in typical amplifier circuits.
We believe students need to be actively engaged in the learning process in order to start thinking about the way amplifier circuits work and start constructing their knowledge from their own quantitative observations. Design problems, either as exercises to be solved or at the laboratory level can be very helpful.
Using in teaching process simulations (e.g., LTspice, Pspice etc.) with changing parameters and examples of applications is a good way to bridge the gap between theoretical-analytical and design-practical.
Students learn when they are actively engaged in their class activities. To integrate theory with practice there will be a series of practical laboratories such as the use of simulations. Simulation tools enable instructors to introduce complex configurations which can be easily tested by the students. Perhaps contrary to common belief, this development calls for a deeper understanding on the part of the students as a prerequisite for smart and efficient use of simulation. Furthermore, analytical comprehension of basic concepts and strategies is indispensable for good circuit design. A proposal for this is assigning problems to students that depart from the classical theoretical motif we used to in the previous years.
The method of teaching students is very important. The teacher should be very patient in teaching and should move slowly from one idea to another one. What is needed is a combination of lectures, tutorials, laboratory exercises (hands-on and simulations) and projects.
Applying the above, we saw that students’ grades significantly improved in the final exam, while the number of students taking the course Final exams increased from 105 to 120 students. Furthermore the number of students who take the optional “Analog VLSI design” course in the semester after “Electronics III” has increased from 15 to 40.
The remainder of this paper is organized as follows. Section II describes the OCTC method principles and outlines some useful analysis tools. It also presents some examples given to the students and some of their misconceptions regarding OCTC. Furthermore, a sampling test given to the students is described. In Section III, the results of the students’ performance in assignments, recent final exams and their answers in the sampling test and two surveys are demonstrated and discussed in Section IV. Finally, Section V concludes this work.
4. Discussion
From the above results (
Table 5 and
Table 6), it can be concluded that the majority of the students managed to correctly solve the problems. Students were able to predict which of the capacitances has the greatest impact on limiting the amplifier bandwidth. However, there were some students who were not able to overcome the difficulties. The wrong calculation of the resistance seen across the terminals of the capacitors was the main mistake of those who answered incorrectly. This was due to deriving an incorrect small-signal equivalent circuit initially, or incorrect equivalent circuit transformations during steps 1–4 of the OCTC method and wrong mathematical calculations at steps 5–6. Students who used the frequent form of
Figure 1 solved the problems easier and ended up with the correct solution. Most of the students who did not use it were confused, arrived at the wrong solutions or did not complete the problem. The last ones were mainly students who missed some (or all) of the OCTC lectures or did not devote the necessary time to prepare.
The undergraduate students’ performance on the homework tasks (the two examples described in Teaching Examples) shows that most of them spend a sufficient amount of time to understand the basic amplitude analysis principles, OCTC method, and its value. The sampling test given to them during the course verifies that about
have an understanding of how to use the analysis tools described in
Section 2.2 in order to compute parasitic capacitance resistors fast. A smaller percentage has shown the ability to estimate which part of the amplifier circuit is responsible for the bandwidth reduction and how this could be optimized [
7,
9,
14,
15].
An observation at the final exam’s performance during the last two years shows important improvement at both the OCTC and overall exam scores. The percentage of students that achieve more than
at the OCTC question has increased from
to
in the February exam and from
to
for the September exam. At the final exam grade, the corresponding increases were from
to
(February) and from
to
(September). Moreover, the percentage of failure has dramatically decreased whereas the average grades have improved by 20–30% for the OCTC question and by
in the February final exam and even
in the September exam. This shows that the students have developed an increased ability in understanding and implementing fundamental amplifier analysis principles, including bandwidth detection, something that undoubtedly will assist them in the direction of analog integrated circuit design [
15].
In the first survey, students’ expectations and interest in the course were moderate at the beginning of the semester (questions Q[1.1], Q[1.2] and Q[1.3]). However, their answers in questions Q[1.4] and Q[1.5] reflected their positive attitude toward the use of simulations. In the second survey, students admitted that the course had increased their ability with the OCTC method (Q[2.1]) and they were interested in the course (Q[2.2]). Of great importance is the question Q[2.3], which roughly measures the ratio between two perceived variables, learning versus required effort. Students’ answers indicated satisfaction with the course. The attitude of the students toward the use of simulations (Q[2.4] and Q[2.5]) was very positive. Finally, at the end of the second survey students were requested to make anonymous written comments (open question), if they wanted, about the teaching method and the learning process. From the 144 students, comments were made by 115 and most of them (63) stated that this teaching method should be applied again in the future for other students. This leads us to believe that the course matched the students’ original expectations and that they were satisfied with the teaching method.
The teaching method for students is very important. The positive results gained from this study confirm the crucial role of the teacher in triggering the conceptual development of his/her students. In summary, the findings of the present study, support the conclusion that the OCTC method can be a valuable tool for teaching design concepts in electronics. This study suggests the kind of teaching materials which could be used and the way lesson plans could be designed in order to teach the OCTC method to undergraduate students as a tool for circuit analysis and amplifier design. It may also be used as a guide for implementing teaching ways for other subjects (eg., feedback, oscillators, noise, et.) in electronics courses. In addition, this work provides a prospect for a future study not with an entire class, but with small groups of students too. It is worth noting that the present study focuses on the use of OCTC method in teaching, and therefore its findings should not be considered as a validation for an integral didactical intervention to the theories and subjects of electronics. In the latter case, to arrive at detailed conclusions would require a different research design and a didactical intervention which would last much longer.